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http://www.archive.org/details/cu31924005003367

SHIP CONSTRUCTION

AND

CALCULATIONS.

WITH

NUMEROUS ILLUSTRATIONS AND EXAMPLES.

FOR THE USE OF OFFICERS OF THE MERCANTILE MARINE, SHIP
SUPERINTENDENTS, DRAUGHTSMEN, ETC.

GEORGE NICOL,

Member of Institution of Naval Architects, Surveyor to Lloyd's Register

of Shipping.

GLASGOW:

JAMES BROWN & SON, 52-56 Darnley Street, Pollokshields, E.

London; SIMPKIN, MARSHALL, HAMILTON, KENT & CO., LTD.

1909.

Preface.

THE rapid advances that have been made in recent years both in design
and in the details of construction of steel ships is the writer's best
apology for placing the present volume before the public. The book
is intended to explain in a simple and practical manner some of the pro-
blems met with in the building and subsequent management afloat of ships,
particularly cargo steamships, and while no claim to special originality is made,
it is hoped that the matter presented will be found up to date. It may
be mentioned that publication has been specially delayed so as to include
reference to Lloyd's latest rules, which differ in certain important respects
from those preceding them, and are "more readily applicable to the changing
conditions of construction."

It is hoped the book will be found useful by officers of the Mercantile
Marine, ship superintendents, draughtsmen, and shipyard apprentices, to all •
of whom a more or less intimate knowledge of naval architecture is essential.
To the officer mariner the subject may now be said to be a compulsory
one, in that those who wish to qualify for the certificate of extra master
must pass an examination in it. But besides this, there are other good
reasons why an officer should know something regarding the construction
and theory of ships. For instance, it would enable him, if called upon,
to act as inspector on behalf of his employers at the building of a new
vessel or the repair of an old one. Or, if his vessel were to receive
sudden damage, calling for immediate temporary repairs, it would give him
confidence in directing his crew in the carrying out of these. In the
management of his vessel afloat, a knowledge of simple theory would assist
him at any time to arrive quickly at satisfactory conditions of draught,
trim and stability, unattainable by mere guess-work or a system of trial
and error. In other ways also such knowledge would prove useful.

The examples chosen for illustration throughout the book have been
selected for their practical interest, and every effort has been exerted to
make the explanations simple.

In conclusion, the author begs to thank Messrs. J. L. Thompson & Sons,
Ltd., Sunderland, for their kind permission to publish diagrams and results of
calculations of vessels built by them ; and he also desires to acknowledge
his indebtedness to Mr. W. Thompson, B.Sc, for help in reading the proof
sheets, and in verifying the examples, as well as for other valuable assistance
rendered while the work was passing through the press.

Glasgow, November, 1909.

CONTENTS.

PAGE

CHAPTER I.
Simple Ship Calculations ........ i

CHAPTER II.
Moments, Centre of Gravity, Centre of Buoyancy .... 25

CHAPTER III.
Outlines of Construction . . .... 42

CHAPTER IV.
Bending Moments, Shearing Forces, Stresses and Strains ... 45

CHAPTER V.
Types of Cargo Steamers ....... 75

CHAPTER VI.
Practical Details .... 93

CHAPTER VII.

Equilibrium of Floating Bodies, Metacentric Stability . . .177

CHAPTER VIII.
Trim . . . . ... 197

CHAPTER IX.
Stability of Shits at Large Angles of Inclination . 217

CHAPTER X.
Rolling . ........ 254

CHAPTER XL
Loading and Ballasting ..... . 272

APPENDICES.
Appendix A .....

297

Appendix B — Table of Natural Tangents, Sines and Cosines; Weights

of Materials ; Rates of Stowage . . 305

Appendix C — Additional Questions . . 309

Index ....... . 324

SHIP CONSTRUCTION AND
CALCULATIONS.

CHAPTER I.
Simple Ship Calculations.

A KNOWLEDGE of the principle of moments and of how to calculate
areas of surfaces having curved boundaries may perhaps be said to
be the only indispensable requisites in dealing with ordinary ship
calculations.

In view of this we propose to spend a little time on these subjects,
first taking up areas of surfaces, and afterwards, as may be found convenient,
the principle of moments, especially as applied to ship problems.

The area of a plane surface bounded by straight or curved lines may
be defined as the number of units of surface contained within its boundaries.
The unit, in English measure, is usually a square foot, although it is some-
times taken as a square inch, and sometimes, although more rarely, as a
square yard. In France, and on the Continent generally, the metrical
system is employed, the units of surface being the square metre and square
centimetre, respectively. These metrical units have many points of advantage,
but as the square foot is more familiar to us, we shall make it the standard
in our calculations.

The simplest figure of which we may obtain the area is a square,
whose chief properties are — all sides equal, and all angles right angles. In
fig. i, A B C D is a square, the length of one side being, say, 6 feet. If
two adjacent sides, such as A B and A D, be divided into 6 equal parts, and
lines be drawn through the points of division parallel to these sides, as
shown in the figure, the square will contain 36 small squares, each of which
has its 4 sides equal to a foot, and encloses one unit of area. There are,
therefore, 36 units in a square having a side of 6 feet. It is obvious that
to find the number of units in any square it is only necessary to multiply
the length in lineal units of one side by itself.

2 SHIP CONSTRUCTION AND CALCULATIONS.

In passing to a rectangle the rule for the area is the same as for a
square, i.e., the length of two adjacent sides are multiplied together, these
being, however, in this case, unequal. As an example, let the adjacent
sides in a rectangle be 16 feet and 8 feet long respectively. By the rule,
we have — Area = 16x8 = 128 square feet.

Fig. h

Fig. 2.

To find the area of a rhomboid, which is a figure having its opposite
sides and angles equal, but none of the angles right angles, a modification
of the previous rule is used. KLMH (fig. 2) is such a figure. From L
and M drop perpendiculars on K l\l, or K N produced, as shown. It is
easy to prove that the rectangle L P Q M = rhomboid L K N M, since obviously
the triangles L K P and M N Q are equal in area. But the area L P Q M =

Fig. 3.

L M x Z. P, from which it follows that to obtain the area of a rhomboid
the length of a side should be multiplied by the perpendicular distance
between it and the one opposite.

This rule is specially useful in explaining the method of obtaining the
area of a triangle. Let ABC (fig. 3), be any triangle. Complete the
parallelogram AGED, and from B drop a perpendicuhr on A G produced.

SIMPLE SHIP CALCULATIONS. 3

Obviously A B bisects the parallelogram A G B D, and, as we have just seen

Area ACBD = ACxBE,

A P v R F

Therefore, the area of the triangle ABO equals By this rule the

area of any triangle may be obtained ; and it is seen that we only require
to know the length of one side and the vertical distance between that side
and the point in which the other two sides intersect. Thus, a triangle having
a base of 25 feet, and a vertical height of 22 feet, will have an area of

25 x 22 .
— 2 75 square feet. The area of any plane figure bounded by straight

lines may be found by one of the foregoing rules, or by a combination of
them, and it should be noted that the earliest rules applied to the finding
of the areas of ship waterplanes and sections were of this nature.

Let ABODE (fig. 4) be a portion of a ship's waterplane. Bisect A E
in F, and through F draw a line perpendicular to A E to intersect the curve
in 0, FO will be parallel to A B and DE. Join B and D by straight
lines, then ABCF and FGDE will be trapezoids. A B, F 0, ED are called
ordinates to the curve : let these lines be represented by the letters y 1 y. 2
and f/ 3 , respectively ; and let h be the common distance between consecutive
ordinates. Obtain now the areas of the trapezoids ABCF and FGDE by ap-

plying the rules already established. Draw B G parallel to A F, meeting C F in
G, then—

B G x G
Area B0G= , and area A B G F = A Fx A B.

Using the symbols, these may be written —

Area B 06 = ^^
2

Combining we get —

x h, and area A B G F = h x y v

Area A B F= — (</ x + y. 2 ).

h
In the same way area FGDE ^ — (ijz + y)

,-. whole area A B D E - — (#1+ 2y 2 + y 3 ).

4 SHIP CONSTRUCTION AND CALCULATIONS.

The rule may be applied to curves with any number of ordinates.
For example, take one having five as in fig. 5.

By the rule, area ABDE = —(y x + 2y 2 + y % )
and, area EDHJ = —(y->+ 2y 4 -\-y 5 )
.\ the whole area, ABDHJ = —{y x + 2y. 2 + a(/ 3 + 2</ 4 + £/ 5 )

This is called the trapezoidal rule for obtaining areas of surfaces bounded
by curved and straight lines. It may be stated as follows : — To obtain the
area of any plane surface, bounded by three straight lines and a curved
line, two of the straight lines being perpendicular to the other, which is
taken as the base of the figure — divide the base between the end ordinates
into any number of equal parts, and through the points of division draw
perpendiculars to the curved line, as in figure 5 ; measure the length
of these ordinates, taking one foot or one inch as the unit of measurement ;
then to the sum of all the ordinates, except the end ordinates, add half the
sum of the end ordinates ; the result, multiplied by the normal distance
between any two ordinates, measured in same unit, will be the area of the
surface, approximately.

Example. — Let length of base = 48 feet. Let there be 5 ordinates, spaced
as directed, giving a common interval of 12 feet, and let the value of the
ordinates be in feet — y 1 = 3 ; y. 2 = 10 ; y 3 = 16 ; y±= 12 ; y 5 — 4, respectively.

Tabulating the information, the calculation becomes —

Ordinates.

Multiplies.

Function of
Ordinates.

IO
16
12

1
I
I

1- 5

lO'O

i6'o

I 2'0
2 O

4i 5

Area approximately = 41/5 x 12 = 498 square feet.

SIMPLE SHIP CALCULATIONS.

The area obtained as above is less than the actual area required by
the small areas enclosed between the straight lines joining the extremities of
each two consecutive ordinates and the curve, as indicated by the hatched
spaces in fig. 5. It is clear that by taking a great number of ordinates,
these hatched areas may be made extremely small, but however numerous
the ordinates may be, the area obtained in this manner is always less than
the actual area. It is obvious, too, that the difference is greatest w hen the
curvature is excessive; this rule, therefore, will give most accurate results
when applied to surfaces having boundaries with comparatively little curvature.
In applying the rule to ordinary ship curves the ordinates should be close
spaced at the points of greatest curvature, and wider spaced elsewhere.

If the curves met with in ship design were of regular form, equations
to them could be deduced by means of which the correct areas of surfaces
enclosed by them up to any point could be written down. Unfortunately,
this is not so. No rigid equation can be applied to ordinary ship curves,
but it is found that no great error is usually involved in treating them as
parabolas,* and this is now the common practice.

* A parabola may be described as the curve forming the line of intersection of a right
cone with a plane parallel to one of its sides ; it is also sometimes defined as the locus of a
point which moves, so that—

its distance from a fixed point

its distance from a fixed straight line

■A.

Fig. 6.

pjr

The fixed point is called the focus and the fixed straight line the directrix. In fig. 6, P
is the parabola, X and Y the co-ordinate axes, M L the directrix which is parallel to Y t
and S the focus, or fixed point. If, now, any point P in the curve be taken, we have —

S P

m p = 1| MP being perpendicular to the directrix. PN, the ordinate to any point P, may be

expressed in terms of the abscissa and a constant, thus —

PN' 2 = C x ON.
If the value of G be varied, other parabolas will be obtained also passing through 0.

SHIP CONSTRUCTION AND CALCULATIONS.

By an important formula, known as Simpson's Rale, named after the
inventor, the area enclosed by any parabolic curve may be obtained. The
rule may be stated as follows :— Simpson's First Rule. — Through the extremities
of any chosen base-line, draw verticals to cut the curve. Divide the distance
between these verticals into an even number of spaces, and through the
points of division draw ordinates to the curve ; which ordinates will therefore
be odd in number. Measure the length of each ordinate, and then add
together four times the sum of the even ordinates and twice the sum of the
odd ordinates, omitting the first and last. To this total add the sum of
the first and last ordinates. Finally, multiply the result by one-third the
length of the common interval between the ordinates ; this will give the
exact area of the surface if the curve be that of a common parabola, and
a close approximation to it if the curve be an ordinary shipshape one.

This rule is of immense value in ship calculations, and we shall pro-
ceed to take a few examples showing its application. Let ABC (fig. 7), be
a ship's half-waterplane of which the area is required. The base A G is
divided as indicated in the rule, and the ordinates y x y 2 y 3 , etc., are drawn

Fig. 7.

through the points of division ; h is the common interval between the ordinates.
We may write—

Area A B = j(y 1 + 4# 2 + 2^4-4^4+ 2^ + 4^+ 2t/ 7 + 4 */ 8 + &)■
It frequently happens that the curvature is greater at the ends of the
waterplane, and a closer approximation to the area is attained by inserting
intermediate ordinates at these places, as shown in fig. 8. The total area
is now made up of three portions, as follows : —
Fig. 8. JL

Area A ED

6 -(0i + 40U + 0s) = ~(ki + 2</H + i# 2 )-

Area DEF6 = j (</ 2 + 4^ + 2</ 4 + 4 </ 5 + 2</ 6 + 4 </ 7 + </ 8 ).
h h

Axe&FGC = j-iyt + Ay&t+y*) = j(ii/ 8 + ays* + £#>)■

Combining these portions, we get —

Area A BG = ~{ly x + 2t/ij + ihy 2 + 4y, + 2</ 4 + 4 </ 5 + 2 y 6 + 4^7 + i£jy a + 2y 9 + Jy 9 ). ( 1)

SIMPLE SHIP CALCULATIONS.

Suppose now that the ordinates have certain definite values beginning
with (/i, as follows: — *i, 5, n"6, 15*4, i6"8, jyo, 16*9, 16*4, 14/5, 9-4, 'i,
and that the total length of the plane is 200 feet: we could find the area
by filling in the values in equation (1), but it is. more convenient to tabulate
the figures, as follows : —

No.h. of
Ordinates.

\ Ordinates.

S.M.

Function of
Ordinates.

2
3

4
5
6

7
8

%l
9

5'o

n-6

i5"4
i6-S
iyo
i6'9
i6"4

i4-5

9"4

1
2

4
2

■05
lO'OO

I7'4
6r6

33' 6

68'o

33-S
65-6

2175
18-8

'°5

33°' 6 5

Area A B = 330*65 x ^^ -755'4 square feet-
3
This result, of course, must be multiplied by 2 for the area of the complete

waterplane.

As showing the application of the rule to transverse sections, take the

following : — The half ordinates in feet of the midship section of a vessel

are — 12-5, 12-8, 13, 13, 12*8, 12*4, ir8, 10*4, 6'8 and -5 respectively; the

common interval is 2 feet ; between the two bottom ordinates an ordinate

at half interval is taken, its value being 4 feet; find the area of complete

section. Arranging the work as before, the calculation becomes —

Nos. of
Ordinates.

-"} Ordinates.

S.M.

Function of
Ordinates.

I
2

4'°
6-8

_1_

'25

8*o

IO"2

10 "4
irS

4
2

41*6
23-6

5
6

I2'4

12*8

49"6
25-6

7
8

13.0
13-0

52-0
26'0

12-8

51'^

12-5

iz'5

300*55

2
Whole area = 300*55 x - x 2

72 square feet.

= 4007:

S SHIP CONSTRUCTION AND CALCULATIONS.

We have seen that to apply Simpson's First Rule to finding the area of
any surface having a curved boundary, there must be an odd number of
ordinates, and not less than three. It is, however, sometimes necessary to
find the area between two consecutive ordinates, as, for instance, between
f/i and y 2 in fig. 9. To do this we employ another rule known as the

Fig. 9

Five Eight Rule, which may be stated thus :—

Five Eight Rule. — Three ordinates being given, to obtain the area between
any two multiply the middle ordinate by 8, the ordinate forming the other
boundary to the space whose area we are finding by 5, and the remaining
ordinate by - 1 ; the algebraic sum of these products, when multiplied by T V
the common interval between the ordinates, will give the area required.
For example —

Area A BCD (fig. 9) = — (Stfi +%*-&)

and area D C £ F = —(5^ + 8z/ 2 -(/ x ).

If these be added we get, after re-arrangement —

Total area A B E F = ™ (</ 1 + 4</a + ffs),

which shows that the Five Eight Rule is based on the same assumption
as the first rule, namely, that the curve is that of a common parabola.

Take a practical example. — The length of the half ordinates of a portion
of a ship's waterplane are in feet, 6, 7 '6 and 8, respectively, the common
interval being 9 feet. Find the area of the portion of the full waterplane
between the first and second ordinates. Tabulating, we get —

£ Ordinates.

S.M.

Function of
Ordinates.

6-o

7-6
8-o

5
8
-1

3°'°
6o-S
-8-o

I
1

82-8

Area = 82*8 x — x 2
12

124*2 square feet.

SIMPLE SHIP CALCULATIONS.

If the area between the second and third ordinates were required, the
calculation would be —

£ Ordinates. S.M.

Function of
Ord.uates.

6'o ! -I
7*6 ! 8

s-o ; 5

6-o
6o.8
40*0

94-3

Area = 94*8 x — x 2 = 142-2 square feet.

Besides the First Rule, which requires an odd number of ordinates for
its application, Simpson introduced another one specially adapted for figures,
having 4, 7, io, 13, etc., ordinates, the number of ordinates to which the
rule applies being given by a general expression. Obviously, this rule will
apply in cases where the first rule would fail, and therein lies its importance.
The statement of the rule is as follows : —

Simpson's Second Rule. — Choose any base line in the surface, and,
through its extremities, draw ordinates to the curve. Divide the space between
these limits into equal parts so as to obtain a number of ordinates as given
by the general expression (3/7 + 4), where ll is zero, or any positive number;
multiply the end ordinates by unity, the 2nd, 3rd, 5th, 6th, 8th, 9th, etc.,
by 3, and the 4th, 7th, 10th, etc., by 2. Add these results together and
multiply the result by f- the common interval between the ordinates. The
quantity thus obtained will be the area of the surface within the given
boundaries very nearly for ordinary ship curves.

Practical Example. — Find the area in square feet of a portion of a water-
plane whose half ordinates are, 2, 6*5, 9*3, 107, n, 11, 10, 7*4, 3^6, and
■2 feet, respectively, the common interval between them being 14 feet.

Arranging the figures as in previous examples, the calculation takes the form —

No. of
Ordinates.

\ Ordinates.

S.M.

Function of
Ordinates.

2*0

2'0

6-5

-1

x 9'5

9*3

27-9

107

21*4

II"0

33^

II'O

33'°

10.0

20'0

7 '4

22"2

3-6

io-S

I9CO

Area of full-plane between the given end ordinates = 190 x 14 x „ x 2 = 1995
square feet.

SHIP CONSTRUCTION AND CALCULATIONS.

Reviewing our rules for areas, we now find that surfaces can be dealt
with having 3, 5, 7, 9, n, 13, etc., ordinates, as required by the First Rule,
and 4, 7, 10, 13, 16, etc., ordinates as required by the Second Rule; for
surfaces having ordinates whose number is not included in the foregoing, such
as those with 6, 8, 12, 14, etc., ordinates, no single rule will apply, and a
combination of the rules given must be resorted to. Take, for instance, a
surface with eight ordinates as shown in figure 10 : —

Examining the figure, we note that the portion A BCD may be treated
by the second rule, and the remainder DGEF by the first. Proceeding
thus, we get —

Area A B C D = g% x + 3</ 2 4- 3^3 + ^)

and area DC E F = ~(y 4 + 41/ 5 + 2y 6 + 4*/ 7 + y B ).

Combining these quantities and re-arranging so as to get a common factor outside
the bracket, we have —

h
Whole area A B EF = -(i^ +3§# 2 + 3§03+ 2S& + 4& + ^ 6 + 4#7 + </ 8 )-

Calculate the area of fig. 10, assuming the ordinates to be 12 feet apart,
and of the following lengths: 167, 24*4, 28-9, 30*3, 29-9, 27-3, 22*3, and
14-9 feet, respectively. The work will be as follows : —

No. of
Ordinates.

Ordinates.

S.M.

Function of
Ordinates.

3
4

5
6

7
8

167
24-4
28-9

3°'3
29.9

2 2'^
14-9

4
1

i8-8

82-35

97*53

64'39
1 19*6

54'6

89"?
I4'9

54i"37

541*37 x 12 x 3
Area A B E F = - b — ~~ = 2436*12 square feet.

It should be noted that the area of the above surface could have been
obtained by combining the First Rule with the Five Eight Rule. The method

SIMPLE SHIP CALCULATIONS.

would be very similar to the above, and we leave the reader to work it out
for himself.

Tcheuychkff's Rule. — This method of finding the areas of plane surfaces
having curved boundaries differs in certain important respects from that of
Simpson's Rule. The ordinates, for instance, are not equally spaced, as in
the latter case, but arbitrarily, according to the number of them employed ;
nor are they treated by multipliers. All that it is necessary to do to obtain
the area in a given case when the ordinates have been placed in position,
is to measure the lengths of the latter, add these together, divide the
sum by the number of the ordinates, and multiply the result by the length
of the base line of the figure.

Fig. 11.

As an example, let us find the area of A B D (fig. 1 1). Here we
have six ordinates spaced according to rule, and numbered i, 2, 3, etc.,
in the sketch ; the total length of the base is 20 feet. Applying the rule,
we have —

No. of
Ordinates.

Ordinates.

I
2
3

5
6

0"77
4*40

5'33
57°

1 -83

23-28

. « « n 2 V28 X 20

Area of A BCD = -^—7

— 7 7 '6 square feet.

So far, the application of the method appears to be simplicity itself;
but a little trouble is introduced in the spacing of the ordinates. This is
done from the middle of the base line as a starting point, and symmetrically
to right and left, the distances being in accordance with the figures obtained
when the half length of the base is multiplied by certain fractions given
in Table 1 .

In obtaining the spacing of the ordinates in the practical example above,
the half length of the base, or 10 feet, was multiplied by '2666, -42 2 2,
and '8662, giving as positions to right and left of the point x, the distances
in feet, 2*666, 4*222 and 8*662; and in the same way, by using the corres-
ponding multipliers, the positions of any other number of ordinates could be

t 2

SHIP CONSTRUCTION AND CALCULATIONS.

determined. It should be noted that where there is an odd number of
ordinates, one occurs at the origin, that is, the middle point of the base
line.

Table i.

Number of Ordinates and Multipliers for Same.

"5773
'5773

•7071

"OOOO

•7071

"7947
■1876
•1876
'7947

■8325

•3745

"OOOO

■3745
■S325

•8662
'4225

•2666
•2666
•4225
•8662

•8839

'5 2 97
'3 2 39

'OOOO

•3239
■5297

.8839

.91 16
"6010
■5288
•1679

"OOOO

■1679

•528S

■6010
•9116

Of course, where a plane is of extended length, it may be necessary
to have more ordinates than is provided by any of the columns of Table 1, in
which case the area might be obtained by repeatedly applying any of the
rules. For instance, if 18 ordinates were required, the two-ordinate rule might
be applied nine times, the three-ordinate rule six times, the six-ordinate three
times, and the nine-ordinate rule twice. This adaptability of the system has
caused some calculators to use only the two-ordinate or three-ordinate rules,
repeatedly applying them as necessary, much in the same way as with
Simpson's 1st and 2nd rules, which, of course, apply initially to figures with
three and four ordinates, respectively.

Fig. 12.

As an illustration of the foregoing, we have again taken the figure
A B D y whose area has been found by directly applying the six-ordinate
rule, have divided it into three parts, and have obtained the area by means
of the two-ordinate rule. Fig. 12 shows the ordinates in their new spacing,
and numbered from left to right The calculation may be arranged as
follows : —

SIMPLE SHIP CALCULATIONS.

No. of
Ordinates.

Ordinates.

I
2

5
6

o*77
4*o4
575
5*37
4 - 9 3
.-83

23-24

2 ^ * 2A X 20

Area A B D = 7 — 77*46 square feet,

which is seen to be very nearly that by previous rule.

It may be mentioned that the area of this figure by Simpson's First
Rule with seven ordinates is 77*03 square feet. Tchebycheffs method is
said to give as accurate results as Simpson's, and with a less number of
ordinates. It has not been found, however, of such universal usefulness in
ship calculations as Simpson's, and, for this reason, the older method is more
generally employed.

VOLUME. — We have seen that the common English units of area are: —
a square inch, a square foot, and a square yard; the corresponding units of
volume are, a cubic inch, a cubic foot, and a cubic yard. While area is
a measure of surface, and therefore deals with two dimensions, volume is a
measure of space, and has to do with three. In fig. 13, the top surface

A B D is assumed to represent an area of one square foot ; the block is
one foot thick, therefore the whole figure A B C G F E A represents the new
unit, viz. : — a cubic foot. In finding the quantity of space or volume that
any object occupies, we merely estimate how many times a standard unit
volume, such as a cubic foot, is contained in it. For example, state in cubic
feet the volume of a rectangular block 25 feet long, 15 feet broad, and 3 J

SHIP CONSTRUCTION AND CALCULATIONS.

feet thick. Let fig. 14 represent this block. The area of the upper surface
A BCD = 15x25 = 375 square feet. If the block were one foot thick, 375

would also measure the capacity in cubic feet ; the actual volume will
obviously be 3! times this quantity, therefore : —

the volume = 375* 3 '5 = 1312*5 cubic feet

From this it appears that to obtain the volume of any rectangular solid,
such as that in fig. 14, it is merely necessary to find the continued product
of the three principal dimensions. There are various rules for obtaining the
volumes of regular solids, and we proceed to state a few of them without,
however, giving the proofs ; these may be obtained by referring to any work
on mensuration.

1. Volume of a pyramid with any form of base = area of base x J height

(perpendicular).

2. Volume of sphere = diameter 3 x - — ~f~~~*

3. Volume of an Ellipsoid = length x breadth x depth x - — p — •

Passing from these, we come next to consider methods of finding the
volumes of solids of more or less irregular form, such, for example, as the
immersed body of a ship.

Fig. 15 shows, roughly, a portion of a ship's body — say below the load
waterplane — which may be supposed represented by A ED FA. Obviously,
we have dealt with no rules which admit of direct application here. In

finding the volume it is sometimes found convenient to proceed as follows :

First, assume the body to be divided by an odd number of equidistant

VOLUME.

transverse planes (nine is shown in the figure), and calculate the areas of
each of these planes from the keel up to the horizontal waterplane AFDEA.

Next, take a horizontal line, HJ fig. 16, having a length equal, on some
scale, to the length of the vessel, and erect equal-spaced ordinates to correspond
with the transverse sections of the body previously mentioned. On each of
these ordinates, which are numbered 1, 2, 3, 4, 5, etc., in fig. 16, measuring
from the base line H J, mark off to scale the number of square feet in the
corresponding section of the vessel. Draw a fair curve through the points
so obtained, and the surface H L J will have an area representing the cubic
capacity of the body.

2 2i2 2 2*3

That the foregoing statement is true may be very simply shown. Let
the space between any two sections, such as 2 and 3, be subdivided by
ordinates drawn through the points 2 1; 2*, 2 3) and where they intersect the
curve, let lines be drawn parallel to H </, as shown. Now, since the
ordinate at 2 represents the area of a section of the vessel at that point,
the little rectangle 2 2 l will represent the volume of a vertical layer between
sections at the points 2 and 2 ± having a constant section equal to area of
vessel at section 2. In the same way the rectangles 2 X 2 2 , 2 2 2 3 , and 2 3 3,

i 6

SHIP CONSTRUCTION AND CALCULATIONS.

will represent volumes of vertical layers, the areas of whose sections will be
those of the vessel at the beginning of each little interval. The sum of the
volumes of the vertical layers represented by 2 2 Xi 2 2 2 2 , etc., between
sections at 2 and 3, will be less than the actual volume of the vessel at
this part, and the deficiency is obviously represented by the areas of the
little triangles between the tops of the rectangles and the curve. But by
making the division close enough, the areas of these little triangles can be
made as small as we please, so that in the limit the volume of the body,
between sections at 2 and 3, will be truly represented by the area of that
portion of fig. 14 enclosed by the curve, the bounding ordinates, and the
base line. Thus it is clear that, as stated above, the total volume of the
body is represented by the area H L J H.

Take a numerical example. — The areas of the vertical transverse sections
of a vessel up to the load line, in square feet, are, respectively, o, 40, 163,
230, 400, 750, 470, 350, 270, 50, and o, and the common interval between
them is 12 feet. Calculate the total immersed volume of the body. It
will be seen that this is merely a question of obtaining the area of a figure
such as H LJ //, and the work may therefore be tabulated as follows : —

No. of ■ /rea of

g ^. Function
of Areas.

3
4

5
6

7
8

9
10

163

230
400

75°
470

35°
270

■ 5°

4
->

4
1

160

326
920
800

3000
940

1400

540
200

8286

Using the sum of the function of areas: volume of vessel = 8->86 x —

1 • r 3

= 33144 cubic feet.
Besides this method of obtaining the volume of a vessel by using the
areas of transverse vertical sections, there is another, and for many purposes,
a more convenient one, which entails the use of the areas of horizontal
sections, or waterplanes. Reverting to fig. 15, the upper plane A ED FA is
such a horizontal section; another one is represented by GKHLG. To fully
take account of the vessel's form, a sufficient number of these horizontal
sections are required. In fig. 17, which shows the midship section of a vessel
the traces of these horizontal planes with the plane of the paper are indicated
as horizontal lines, numbered 1, 2, 3, etc. Only one half of the body is

VOLUME.

shown, the vessel being symmetrical about the middle line plane. The area
of each of the horizontal planes is first calculated and set off to some scale,
to the right of the middle line 5/3, on a horizontal line opposite the water-
plane to which it refers. In fig. 1 7 these areas are represented by B
for the first waterplane, FG 1 for the second, and so on. A fair curve GG^D,
drawn through these points will obviously, from our previous consideration,
enclose an area representing the volume of the vessel from the keel to the
first waterplane; and therefore, to obtain the immersed volume of the vessel, it
is only necessary to calculate the area of BCD.

Fig. 17.

W.P.

ew.E

3 W.P.

4W.P.

Take one numerical example: If the areas of the waterplancs of the vessel
in fig. 17, beginning at the upper one, be 8000, 7600, 7000, 6000, 4500, 2800,
and 150 square feet, respectively, and the distance between them be 3 feet, what
will be the total volume? The work of finding this area we tabulate as
follows : —

No. of
W.P.

A'ea of
W. P.

S.M.

Funct 011s
of Are s.

SOOO

80OO

7600

304OO

7000

I40OO

6000

24OOO

5
6

4500

280O

15°

6750
560O

8S825

volume below No 1 waterplane = 88825 x = 88825 cubic f" ee t-

1 8 SHIP CONSTRUCTION AND CALCULATIONS.

DISPLACEMENT AND BUOYANCY.— At this point, for a reason which
shall appear presently, we must endeavour to explain an important hydrostatic
principle known as the Law of Archimedes. This law asserts that if any
body be immersed in a fluid it will be pressed upwards by a force equal
to the weight of the volume of the fluid which it displaces ; and if the
body float at the surface of the fluid with only a portion of its bulk im
mersed, that the volume of fluid displaced will have the same weight as the
total weight of the body. Thus, a box-shaped vessel, ioo'x2o'xio', floating
in salt water, with half its depth of 10 feet immersed, will displace —
ioo x 20 x 5 = 10000 cubic feet of the fluid.

And since we know, or may easily verify by experiment, that a cubic foot
of salt water weighs 1025 ozs., or 64 lbs., and therefore that a ton of salt water

2240 '
occupies a space of -7 — = 35 cubic feet, we are able to write —

Weight of water displaced by vessel = ■ = 28 5*7 tons -

By the Law of Archimedes this weight is equal to that of the vessel and its contents.
The following is a simple proof of this important principle. If the body

Fig. 18

represented by A in fig. 18 be placed in a fluid of greater specific gravity
than itself, it will float with a part of its bulk above the surface as shown.
The immersed portion will be pressed in every direction by the fluid, those
pressures which act on a section parallel to the plane of the paper being
indicated by arrows. If, now, we imagine the fluid surrounding the body
to become solidified, and the body itself to be non-existent, a cavity will
remain having the exact shape of the immersed form of the body. If
finally, this cavity be supposed filled to the top with the same fluid and
the surrounding solidified fluid be supposed to .return to its former state
there will be a free level surface, and consequently the equilibrium will not
be disturbed — that is to say, the fluid occupying the cavity will have the same
statical effect as the body itself, since the same resultant upward pressure
keeps each of them in equilibrium. From this it at once follows that the
weight of the floating body is the same as that of a volume of the fluid
occupying a space equal to that of the immersed portion of the body. This
principle is of enormous value to the naval architect, for by it, when a vessel

DISPLACEMENT AND BUOYANCY.

is floated, he knows that its weight, including contents, is equal to that of
the displaced water. He has thus an infallible means of checking his calcula-
tions and of forming a basis on which to estimate the amount of cargo the
vessel will carry.

We now see the importance of being able to calculate correctly the
volume of the immersed body of a ship. We have described two ways in
which this work can be done, and pointed out that the method involving
the use of horizontal areas is preferable to the other, because of its greater
convenience. This is seen, for instance, in the ready means which it affords
of obtaining the volume, and therefore the weight of the displaced water at
each of the various waterplanes indicated on fig. 17. These intermediate
displacements, although not of special value of themselves, when plotted to
scale at corresponding draughts, give a curve from which the displacement
at any draught up to the load-line may be read off. This curve constitutes
what is known as the displacement diagram.

As a practical example let us construct such a diagram in a specific case. Con-
sider the vessel whose waterplane areas were used in the example on page 17. The
volume up to the load waterplane was there determined to be 88,825 cubic feet.*

Dividing this by 35 we obtain

>25

= 2538 tons as the ordinate of the displace-

ment curve at a draught of 15 feet. Referring now to fig. 17, the volume
up to the 2nd waterplane, or to a draught of 12 feet, is represented by
the area DFG y . The simplest way of obtaining this volume is to deduct
the layer between the 1st and 2nd plane represented by the area BFG l C
from the total volume. The displacement to the 3rd waterplane may be
found by direct application of the 1st rule, while for the value to the 4th
waterplane, the volume between the 1st and 4th planes should be got by
Simpson's 2nd rule, and the result deducted from the total volume. The 1st
rule will be suitable for finding the displacement to the 5th plane, the half
interval being used in this case. In the following table we show these cal-
culations carried out as suggested ; the final results are arranged by themselves
for easy reference : —

Displacement Calculation.

No. of
Sect.

Areas of
W. Planes

S.M.

Function
of Areas.

S.M.

Function
of Areas.

S.M.

Function
of Areas.

S.M.

Function

of Areas.

S.M.

Function
of Areas.

I
2

3
4
5,

SOOO
7600
700O
6000
4500
2803
150

4
2

i\
2

8000

3040O

I400O

24000

6750

5600

4
2

7000

24003

6750

5X0

2
\

2250
5600

- 1

40000
60S00
- 7000

3
3
1

8000
22S00
21000

600O

S8825

43425

7925

93S00

57SOO

1 There is assumed to be no displacement below the 6th waterplane.

20 SHIP CONSTRUCTION AND CALCULATIONS

Displacement to load waterplane

Displacement of layer between ist and 2nd W.P.
Displacement to 2nd waterplane
Displacement to 3rd waterplane

Displacement of layer between ist and 4th \V. P.
Displacement to 4th waterplane
Displacement to 5th waterplane

SSS25 x 3 Q .
3 — 2 = 2^S tons.

35 x 3

93800 x 3

35 x I2

670 tons.
1S68 tons.
1240 tons.

^ 43425 x 3

35 x 3

= 578oox3X3 = l8 8tonS-
35x3 b

— 680 tons.

226 tons.

= 79 2 5 x 3
35 x 3

The construction of the displacement curve is now an easy matter. Take
a vertical scale of draught A B (fig. 19), and let the distance from A to B equal
15 feet; divide it into five equal parts and draw horizontal lines from the points
of division. Number these horizontal lines from B downwards. Now measure
along these lines, to some convenient scale, distances representing the displace-
ments corresponding to these draughts. A fair curve drawn through the points
will give the diagram required.

Fig. 19.
SCALE OF DISPLACEMENT IN TONS

J5_

, 5 f°, , , ,s c

) I5C0

2000

25,00

1ft

r«¥L

1—

A"WL_

_5_'"iV.L/

tn
^—

■<£
ce
O

_L_

To complete it, a scale of tons should be shown along the top, and the
distance between A and B divided off into feet and inches, so that any draught
can be located immediately. The construction lines 1, 2, 3, etc., being no longer
required, should be erased. It is necessary to note that in reading from the
diagram mean draughts only should be used. For instance, if the displacement
of the above vessel were required when floating at 12 feet aft and 9 feet forward

DISPLACEMENT AND BUOYANCY. 2 1

the draught to be taken on the scale should be = io£ feet.* A horizontal

line drawn out at this draught would intersect the curve in a point showing on
the scale of tons a displacement of 1530 tons. In the same way the displace-
ment at any mean draught, provided it did not exceed 15 feet, could be found.

From the displacement diagram another very useful one may be constructed,
called a "scale of deadweight." It is specially constructed for the use of ships'
officers and others who may have to do with loading operations. It exhibits
in graphic form the weight of cargo put aboard as the vessel sinks in the water,
and may be looked upon as a kind of loading meter by which the officer is able
to tell, at any moment during loading operations, the amount of cargo he has got
aboard, and the amount still to be dealt with to bring the vessel to her
assigned load-line.

In fig. 20 we give an illustration of such a diagram deduced from fig. 19.
It will be observed to consist of two columns, one of which is a scale of draughts
in feet, while the other indicates the amount of immersion caused in the vessel
by the addition of each 2007 tons in her load. The effect on the draught of
quantities less than 200 tons is, of course, found by interpolation.

Fig. 20.

Deadweight

Scale.

Freeboard.

Mean
Draught.

Dead-
weight.

MAIN DECK

LOAD

1600

MOO

1200

G i _JL

7 _ _J1_

1000

800

600

400

200 -

LIGHT

Draught (2538 tons displacement,
1738 tons deadweight).

Draught (Weight of ship including
machinery, 800 tons).

As an example, suppose that ihe vessel, whose deadweight scale is illustrated
by fig. 20, is observed at a certain time during Joading operations to be floating

* This is not absolutely correct, see Appendix A.

f In ordinary cases increments of 100 tons deadweight are indicated.

2 2 SHIP CONSTRUCTION AND CALCULATIONS.

at 7 feet forward and 10 feet 4 inches aft, and that it is required to ascertain
how much cargo has been put on board, and how much has still to be shipped
to sink her to a mean draught of 15 feet?

Mean draught at time of observation = — '■ — - — - = 8 ft. 8 ins.

At this draught there will be 380 tons aboard.

At 15 feet mean draught the vessel will carry 1738 tons, therefore the
amount of cargo still to be shipped = 1736 - 380 = 1356 tons.

CURVE OF TONS PER INCH OF IMMERSION.— Sometimes it is de-
sirable to know how much the draught of a vessel would be affected by shipping
or discharging a moderate quantity of cargo. If the mean draught were known,
this information could be obtained from the displacement diagram or the dead-
weight scale ; it can, however, be more conveniently got by means of a special
diagram, called a "Curve of tons per inch of Immersion," which shows graphically
the number of tons required to sink or lighten the vessel one inch at any
draught. The weight of cargo shipped, divided by a number read from the
diagram, will give the number of inches by which the draught has been altered.

Fig. 27.

TONS PER INCH IMMERSION

| 2 4 6 8 10 tt 14 16 r| |

^^^

_^~-~""

If A be the area of any waterplane, then the weight of a layer of salt

A x ~ A

water one inch thick will be ^ ^ tons ; which by Archimedes' nrin-

35 4 2 ° ^

ciple will also equal the number of tons of cargo necessary to sink or lighten the
vessel one inch at this draught.

To construct the diagram required take any vertical line representing to
scale the full mean draught of the vessel, and at the heights of the waterplanes

CURVE OF TONS PER INCH OF IMMERSION. 23

of which the areas are known, draw horizontal lines. Mark off to scale on each

of these lines the corresponding quantities , and draw a fair curve through

420

the points so obtained. This will be the curve of tons per inch of immersion.

To complete the diagram, as shown in fig. 21, a scale of draughts and of tons

must be drawn and the construction lines erased.

Example. — If the vessel, whose diagram is given alDove, were floating at
a mean draught of 9 feet, what would be the increased immersion due to
shipping 50 tons of cargo? From the curve at 9 feet draught the tons per
inch is found to be i6 - 6, therefore : —

additional immersion — JL— = ^- i inches.
16*6 °

QUESTIONS ON CHAPTER I.

1. State the Trapezoidal Rule for finding areas of plane surfaces having curved boun-
daries, and point out wherein it is inaccurate. The half ordinates in feet of the load water-
plane of a vessel are, commencing from aft, 2, 6*5, 9/3, 107, n, n, 10, 7*4, 3*6, and *2,
and the common interval between them is 15 feet. Find the area of the plane by using the
Trapezoidal Rule.

Ans, — 21 18 square feet.

2. What are the advantages of Simpson's First Rule for finding plane areas, and for what
curve is the Rule accurate? What are the conditions as to the number and spacing of ordi-
nates? The semi-ordinates of the waterplane of a vessel in feet are, respectively, 'I, 5, 11 '6,
15*4, i6'8, 17, i6'9, l6'4, 14*5, 9"4, and *l. The spacing of the ordinates is n feet, find
the area of plane in square yards.

Ans. — 303*6.

3. Given the values of three consecutive and equally spaced ordinates and the common
distance between them, what Rule would you employ to find the area between the first and
second ordinates? If the ordinates in feet are 5, 11 "6 and 15*4, and their spacing n feet,
find the area between the first two.

Ans. — 93*86 square feet.

4. State Simpson's Second Rule. To what class of curve does it apply accurately?
Given *i, 2*6, 5, and 8*3 as the value in feet of the half ordinates of a portion of a ship's
waterplane, and 9 feet as the common distance between them, calculate the area including both
sides.

Ans, — 210*6 square feet.

5. Why are half ordinates sometimes introduced at the ends of plane figures? Deduce the
modification in the multipliers of Simpson's First Rule due to the introduction of a half
ordinate.

6. What are the main points of difference between Tchebycheff's Rule and Simpson's
First Rule for finding plane areas? Compare by an actual practical example the results obtained
by applying Tchebycheff's two ordinate Rule and Simpson's First Rule.

24 SHIP CONSTRUCTION AND CALCULATIONS.

7. Given the areas to the L.W. P. of the transverse vertical sections of a vessel, show
that the volume of the displacement may be expressed as a plane area. If the tranverse
vertical sections in a particular vessel are 4, IOO, 1S0, 240, 260, 242, 190, 120 and 8 square
feet, and the common interval is 15 feet, calculate the volume of displacement.

Ans. — 20,400 cubic feet.

S. Explain why it is preferable to employ the areas of horizontal sections or waterplanes
rather than of transverse vertical sections in calculating the volume of displacement. The areas
of the waterplanes of a vessel are £000, 6000, 4S00, 3600, 2400, 1200, and IOO square feet;
the common interval between the waterplanes is 2 feet. Calculate the displacement in tons (salt
water), neglecting the portion below the lowest plane.

Ans. — 1213-3.

9. What is the "Law of Archimedes"? Explain in what way this Law is important to

the naval architect.

10. Referring to the latter part of question No. 8, calculate the displacement to the various
waterplanes, and plut the diagram of displacement.

11. How is a curve of "Tons per inch of Immersion" constructed? What use is made
of such a curve? The areas of a ship's L.W.P. is 4000 square feet, and the areas of other
parallel water sections are, respectively, 3650, 32:0, 2550, and 24 square feet. The vertical
distance between the sections is 2 ft. 9 ins. Construct the curve of "Tons per inch of Im-
mersion?"

CHAPTER II.

Moments, Centre of Gravity, Centre of Buoyancy.

MOMENTS. — If two equal weights be placed one at each end of a weight-
less lever A B (fig. 22), it is obvious that the point at which they may be

Fig. 22.

supported in equilibrium lies midway between them. If the weights be un-
equal the balancing point will not be at the middle but at some other
position nearer the larger weight.

In books on elementary mechanics it is shown that in all such cases the

P PR

exact position of C may be obtained from the relation (fig. 23), n = irn (i),

Fig. 23.

where P and Q are the unequal weights, and A and OB the distances of the
points of application of the weights from the fulcrum. Cross multiplying,
this equation becomes P x A = Q, C B, (1).

It appears then, that from a consideration of a balanced system of two
parallel forces acting on a rigid bar assumed to be weightless, two items of
interest may be deduced : first, that the position of the point of support must

SHIP CONSTRUCTION AND CALCULATIONS.

be fixed by equation (i); and second, that the moment of the force, or turning
effort, about the point of support on one side must be equal and opposite to
that on the other, as indicated by equation (2).

Thus, if weights of 8 and 12 lbs. be suspended at A and B (fig. 24), the

Fig. 24.

extreme points of a weightless lever 36 inches long, and if X be the distance
of the balancing point from A, we have, using equation (1), — —

from which we get X = 2 if inches.

That the point thus determined by X is the one required is proved by
the fact that the moments of the weights about this point are equal, i.e.,

2i| x 8 = 14^ x 12.

The following is an important theorem ; —

The moment of the re$idta?it of a system of parallel forces in one plane
acting on a rigid body about any point in the plane is equal to the sum oj
the moments of the component forces about the same point. Take the simple
case of two forces acting in the same direction, as in fig. 25. Let A and B

Fig. 25.
A c ft

v
P

be the points of application of the forces ; join A B and assume the line to
be horizontal. Let be the point about which the moments are to be taken,
and R the resultant of the two forces, which may be called P and Q. Drop
a perpendicular from upon the line of action of the forces, which for
simplicity are assumed to be vertical, cutting them in the points D, E, and F
as shown.

It is clear that the above theorem will hold if RxOE = PxOD +
Q x OF, that is, if (P+Q)0E = P(0E -DE) + Q (0 E + EF) or, multiplying
out and cancelling like terms, if P x D E = Q x £ F.

MOMENTS.

Since A B is parallel to D F, this may be written PxAC = QxCB.
But this relation we know to be true, therefore so must be the above theorem.

The theorem will, of course, hold if there be any number of forces acting,
for if the line of action of the resultant be found, the forces acting on either
side of this line may be represented by a single force, and this will reduce the
case to the one just proved.

We are now able to deal with questions in which it is necessary to find
the position of the resultant of parallel forces. Consider the forces P ly P 2i P&
etc., shown in fig. 26, which, in the first instance, we shall suppose acting in

Fig. 26.

\
P

' *
1

>
' R

one plane. In order to determine the position of the resultant we must take
moments about some point in the plane. Drop a perpendicular upon the lines
of action of the forces, which, as before, we assume to be vertical, cutting them
in the points A, B, G, D, and £, and take the moments about a point in this
line, say where it intersects the line of action of the force P v Assuming
the resultant R to act at a distance x from A, we have by the theorem — ■
Rxx = P 1 xo + P 2 xAB + P s xA6 + P 4 xAD + P 5 xAE, from which, since
R equals the sum of the several weights,

_ P 1 xo-[-P 2 xAB + P 3 xA0 + P 4 xAD + P 5 xAE

X " P 1 + P 2 + P3 + Pi+ P 5

Suppose that the forces P Xi P 2i P 3l P 4 , and P 5i are of the following magni-
tudes, viz. — 4, 8, 6, 12 and 10 units respectively, and that the distance A E,
which is 12 feet, is divided into equal parts by the lines of action of the forces,
the distance of the resultant from A will be —

X =

4X0 + 8x3 + 6x6+ 12 X9+ IO X 12
40

7 '2 feet.

If the forces in fig. 26 do not act in one plane, in order to find the
centre of the system it will be necessary to determine its perpendicular dis-
tances from two vertical planes at right angles. If we suppose one of these

2o SHIP CONSTRUCTION AND CALCULATIONS.

planes to be perpendicular to the plane of the paper, anJ its vertical trace to
be represented by the line of the force P lt by a simple moment calculation
about this plane we will determine, not the position of the resultant, but only
of the vertical plane containing it parallel to the plane chosen as the axis.
To fully determine the position we must now find another plane also
containing the resultant parallel to the plane of the paper. Clearly, since it is in
both planes, it must coincide with the line of their intersection. Let the forces
in fig. 26 act at the distances shown from the plane containing P l normal to
the plane of the paper; 7-2 will be the distance from the axis of one of the
planes containing the resultant. To find the other one we must know the
position of each of the forces from a plane parallel to the plane of the paper.
Let these be given by the normal distances y u y<&-y& y* - {js, eacn one having
the same suffix as the force to which it refers. Calling Y the distance of the
plane of the resultant from the axis plane, we have —

Y _ fill + fttfg ~ gsj/g + P*y*- P 5 y 5 •

p x + p, + P 3 + Pi + P 9

Putting in the numerical values 2, 4, - 7, 9, - 5, for y 17 </ 2 , etc., this becomes —
4x2 + 8x4-6x7 + 12 X9-5X 10

40 " T 4

The two values, X ~ 7*2 feet and Y = 1*4 feet, determine the line of the
resultant of the system of the assumed parallel forces.

The preceding principle admits of many important applications, not the
least of which is that to the finding of centres of gravity, to uhich we must
now turn. We begin with a general definition.

CENTRE OF GRAVITY.— If the mass of any body be supposed divided
into an infinite number of parts, the forces or weight due to the attraction of
the earth, acting on the various parts, will form a system of parallel forces of
which the total weight of the body is the resultant • and the point through
which the line of action of this resultant always passes, whatever be the position
of the body with reference to the earth, is called the centre of gravity of the
body.

The centre of gravity of a body is also sometimes defined briefly as the
point at which the weight of the body may be taken to act, no matter what
position it may occupy. Thus, in the case of a ship and cargo, the total
weight is taken as acting at a fixed point when making stability and other
calculations.

It is frequently necessary in dealing with ship calculations to obtain the
centre of gravity of an area. In approaching such questions it is usual to keep
the idea of weight and to consider the area as consisting of a homogeneous
lamina of uniform but infinitely small thickness. Thus, the centre of gravity of
a lamina of circular form is at its geometric centre, as evidently the resultant of
all the forces due to the weight of the various portions of the lamina must pass
through that point. Also the centre of gravity of a lamina of square form is in
the point of intersection of the two diagonals. To find the centre of gravity

CENTRE OF GRAVITY.

of a triangular lamina such as A B C (fig. 27), we may proceed as follows: —
First, bisect A C in D and join B D. This line contains the centre of gravity
of all strips of the lamina parallel to A G, consequently the centre of gravity of
the triangle must be somewhere in it. Next, bisect one of the other sides, say

B % in £", and join A E. The centre of gravity of the lamina must obviously
also be in A E ; therefore, it must be at the point G where the lines A E and
B D intersect. G is at a point one-third of BD from D.

In passing to the case of a lamina having a curved boundary, such, for
instance, as the half waterplane of a ship, we cannot determine the centre of
gravity by such geometrical methods, owing to the irregularity of the form.
The usual practice is in effect to divide the lamina into an infinite number of
elements, to take the moments of these elements about any two axes chosen at

right angles in the plane of the lamina, and to divide the sum of each series of
moments by the total weight of the lamina, each quotient being the distance of
a line containing the centre of gravity parallel to its corresponding axis, and the
centre of gravity itself, the intersection of the two lines. By employing Simpson's

3°

SHIP CONSTRUCTION AND CALCULATIONS.

Rules it is only necessary to deal with specimen elements, which greatly simpli-
fies the work. As an illustration consider the half waterplane A B G (fig. 28).
Divide A B into a number of parts as shown, and draw ordinates to the curve.
Now take a strip of the lamina of very small breadth a at ordinate 5, say ; its
area will be y 5 a, and this may also stand for the weight since the lamina is
homogeneous and of uniform thickness.

The moment of this little area about A B as axis will be —

y*a-

= yi. a

With a as base, set down^- as an ordinate below A B, and draw in the

2
little rectangle shown in the diagram, which will represent the moment of the strip
of area at y 5 . In the same way obtain and plot the moments of elementary areas
at #i> #2, etc. A curve through the extremities of these little rectangles will en-
close an area A D B, which will represent the moment of the area AGB about
the line A B, and consequently, by the principle of moments, the distance of
the centre of gravity of this area from the chosen axis will be given by —
Total Moment Area about A B Area AD B
Total Area ' ° r Area AGB'

As a numerical example, let the above half waterplane be 140 feet long,
and let 11 ordinates be taken so as to suit the application of Simpson's First
Rule to the finding of the areas A D B and A B.

The figures of this calculation are best arranged in tabular form as shown
below. In the first two columns are the numbers of the ordinates and their
values in feet, respectively. The third column gives Simpson's multipliers, and
the fourth the corresponding functions obtained when the ordinates are treated
by these multipliers. It will be seen that so far the work is simply in the
direction of finding the area AGB. The next two columns are for obtaining
the area enclosed by the moment curve, the fifth giving the squares of
the ordinates, and the sixth the functions of the same when affected by the
multipliers.

■ No', of
Ordinates.

Ordinates (ft).

S.M.

Function of
Ordinates.

Ordinates 2 .

Function of
Squares.

I
2

3
4

5
6

7
8

9
10
11

2'5

6"5
9'3

10-7

J I'O
II'O
IO'O

7 4
3"6

4
2

4
1

IO'O

13-0

37*2

2 1*4

44'°

22'0
40'0
14*8
I4-4
*2

•09
6*25

42-25

86-49
114-49

I2I-00

I 21 'OO

lOO'OO

54-76

12 96

-04

•09
25-00
84-50

345'96
228-98
484-00
242-00

4oo - oo
109-52

51-84
■04

2i7'3

r 97i'93

CENTRE OF GRAVITY.

3 1

Using these figures we obtain at once —
Distance of centre of gravity of half \ Area enclosed b y moment curve

plane from axis A B

Area of half plane

*97i'93 x £4
2 3 c

217-3 x y

4*53 feet

It should be noticed that in the calculation the whole squares are employed
in the table, the division by 2 being done at the end, as shown.

We have, as a result of the preceding calculation, fixed the position of
one line containing the centre of gravity. We must now, as already men-
tioned, determine the position of another line also containing it at right angles
to this one. The principle of moments is again employed, and in fixing upon
an axis for the purpose, it is usual to choose an ordinate about the middle
of the plane, as it will obviously mean a less laborious calculation than if
the axis were taken, say, at either end. Care must also be taken .to select nr

Fig. 29.

ordinate which will allow of Simpson's First Rule being applied in arriving
at the areas enclosed by the moment curves. In the present instance either
ordinate 5 or 7 might be employed; the middle ordinate No. 6 is unsuitable
(See fig. 29)

Taking No. 5 as axis, the moment of a small strip at ordinate No.
4 will be y 4 Cth, h being the common interval between the ordinates and a
the breadth of the strip. At ordinate No. 3, the moment of a strip will be
y 3 CtX2h, and so on for strips at the other ordinates, the little area in each
case being multiplied by the number of times of the common interval it is
removed from the chosen axis. The process is repeated for the area on the
other side of the axis, the moment of a strip at No. 6 ordinate being y^a h ;
that for strip at No. 7, */ 7 a 2/7, and so on. To construct the moment diagram,
the moments thus found of the small areas are, as in the previous case, set
down as little rectangles, each on a base a, on the other side of A B at the
points to which they refer, and fair curves drawn as A H F and FKB in the
figure. Evidently, the centre of gravity will be on that side of the axis which

SHIP CONSTRUCTION AND CALCULATIONS.

has the greater moment ; and its distance from the axis will be obtained by
dividing the difference of the moments by the area of the half plane ; thus—

Distance of centre of gravity from axis! Area FKB - Area AHF

through ordinate No. 5, / ~ Area A B

The work of finding the above areas is arranged below. The first four
columns are the same as before; in the fifth are the multipliers representing
the number of intervals each ordinate is distant from the axis through No. 5;
the sixth column gives the functions of the ordinates after treatment by these
multipliers as well as those of Simpson's Rule.

No. of
Ordinates-

Ordinates.

S.M.

Function of
Oidmares.

Mult, for
Leverage.

Function of
.Moments.

2
3

. 4
5
6

9
10
11

*3
2*5

6-5

9*3

10*7

II'O
I I'O

IO'O

7 '4
3-6

4
2

4
1

'3
IO'O

130

37'2
21-4
44-0

22'0
40*0
I4-8
14-4

4
3
2
1

1
2
3
4
5
6

1 '2

30-0
26'0

37'2

94 '4

44-0
44-0

I20"0

59' 2
72-0

I"2

2173

34°"4

Centre of gravity from axis through "\ (34° 4
ordinate No. 5, /

H 4) — x 14

i + ~ ~ = 15-85 f eet.

217*3 x

The centre of gravity of the half plane is therefore situated at a. point 15-85
feet forward of No. 5 ordinate, and 4-53 feet out from the centre line A B.
Obviously, the centre of gravity of I he whole plane, since both sides are alike,
will be in the middle line and at the same distance forward of the axis
namely, 15*85 feet.

Fig. 30.
£ e.

The same useful principle of moments is employed if we wish to find
the centre of gravity of a portion only of a ship's watetplane, say, of the area
ACEB (fig. 30), omitting the space GFLE As before, moments of elemen-
tary areas are taken about A B and about an axis at right angles to it omitting

CENTRE OF GRAVITY.

the portion of the area, CFLE. The resulting distances obtained by dividing
the sum of each of these systems of moments by the reduced area will deter-
mine the position of the centre of gravity of the partial plane from the chosen
axes.

CENTRE OF BUOYANCY.— It was shown, when treating of displacement
and buoyancy, that the weight of any floating body is supported by the
upward pressure due to the buoyancy of the water. Fig. 31 represents in
section a ship floating freely and at rest in still water, and indicates the water
pressures acting on her. It is the resultant of the vertical components of
these pressures, which act everywhere normal to the surface in contact, which
supports, and is therefore equal to, the total weight of the vessel. It now
becomes necessary to state further that the line of action of this resultant,
whatever be the position of the vessel, always passes through a certain point,
viz., the centre of the immersed bulk, or the centre of gravity of the water
that would occupy the same space. This point is called the centre of buoy-

Hg. 31
I

ancy. We now proceed to show how this centre may be determined in any
given case.

In a vessel of simple box-shape, floating at a level waterplane, the point
will obviously be at mid length in the centre line plane, and at a distance
below the surface equal to half the draught. In one of constant triangular
section floating with a side parallel to the surface it will also be at mid length,
but at J the depth below the surface. In a cylindrical vessel floating at even
keel, it will as before be at mid length and at the same distance below the
surface as the centre of gravity of the transverse section. In all these cases,
the conditions being given, the point required can be easily determined. In
ship-shape bodies, however, owing to the irregularity of form, no such simple
methods can be applied. We must, therefore, resort to moment calculations,
just as we had to do when finding areas of surfaces enclosed by ship curves,
c

3HIP CONSTRUCTION AND CALCULATIONS.

Take, as example, a vessel of ordinary form floating at a draught parallel to the
keel-line (see fig. 32). In setting out to find the centre of buoyancy, we ob-
serve, in the first place, that, since the vessel is symmetrical about the middle
line plane, the point required must be somewhere in that plane, and that it
will be fully determined if we know its position relative to a vertical and a
horizontal line in the plane.

To obtain the vertical position of the point, the immersed body is assumed
divided into an infinite number of horizontal layers, and a calculation of moments
made with respect to some horizontal plane, such as that of the load-water line.

Fig. 32.

For the horizontal position, the displacement is supposed divided into transverse
vertical layers and another calculation of moments made ; in this case, with
respect to a transverse vertical plane, such as that of the after perpendicular, or
of a transverse section in the vicinity of amidships. It is only necessary to
correctly plot the results of these calculations in the middle-line plane to obtain
the position of the centre of buoyancy. In practice, as in the case of calcula-
tions of areas and volumes, by using Simpson's or TchebychefFs rules, only
specimen layers of displacement need be dealt with.

Fig.

33.

A x ./|

h\-

A32/I.

^ Ait-

A43A.

kV* 5

As Ah.

A«Ath. y

Turning to fig. 32 we note that between the upper waterplane W.L. and
the keel, four others are introduced at equal distances apart, with, in addition
an intermediate one between the keel and the plane marked No. c. The
plane at half interval is introduced owing to the increased curvature of the
ship's form at that part, which makes a closer spacing necessary to obtain
accurate results.

CENTRE OF BUOYANCY.

Let Au Aj, A 3i etc., be the areas of the waterplanes, h the common in-
terval between them, and a. a very small thickness of layer taken at each
waterplane. The moments of the elementary layers about W.L. will be —
0, A 2 ah, A 3 a 2h, and so on, volumes being treated as weights, the density
being constant. Now take any vertical scale of draughts (fig. 33) and mark
off horizontally at the planes, 2, 3, 4, etc., the corresponding moments just
found, which should be plotted as rectangles of breadths, A 2 h,A 3 2h i etc., and
depths a. A fair curve through the extremities of the little rectangles thus
obtained, starting from the point £, will enclose an area representing the total
moment of the volume below the upper waterplane. If, on the other side of
the axis E /?, another diagram be plotted, the ordinates at the various water-
planes being the corresponding waterplane areas, the area' of this diagram will
represent the total volume of the vessel below No. 1 waterplane.

From our previous considerations it is clear that we may write : —
Distance of centre of buoyancy) __ Area E C B
below No. 1 waterplane, / _ Area ED B

As a numerical example, suppose the areas of the waterplanes in square
feet are, beginning from the upper one, 8000, 7600, 7000, 6000, 4500, 1800,
and 100, respectively, and that the common distance between them is 3 ft.,
with a half interval at the lower end.

In obtaining the vertical distance of the centre of buoyancy below the
upper waterplane, in this, and all similar examples, it is convenient to arrange
the work in tabular form, as shown below.

In the second, third, and fouith columns are the areas, Simpson's multi-
pliers, and functions of areas, respectively. In the fifth column are the
multipliers for leverage, and in the sixth, the products of the lever multiples
and the area functions. The process is seen to be simply that of obtaining
areas such as E G B and E D B by Simpson's Rule.

No. of
Ordinates.

Areas of
Planes.

S.M.

Function
of Areas.

Levers.
O

Function of
Moments.

8000

80OO

-■»

7600

30400

7000

I4OOO

28000

6000

24OOO

72000

5
5*

4500
1800

6750
3600

4
4i

27000
16200

IOO

5°

250
173850

86800

173850 x £ x 3
Distance of centre of buoyancy\ 3

below No. 1 W.P., l~

6 feet.

86800 x 3
3
Volume of displacement below No. 1 W.P. = 86800 cubic feet

SHIP CONSTRUCTION AND CALCULATIONS.

The position of the centre of buoyancy below any ot the other vvaterplanes

may now be obtained. Reverting to fig. 33 — area _ _ !* .. gives the distance

area u t H N

of the centre of the layer between the 1st and 2nd waterplanes from the 1st
waterplane, and by a simple moment calculation, the fall in the centre of buoy-
ancy consequent on the vessel rising to the 2nd waterplane is derived. In the
same way, by first finding the centre of the layer between the 1st and 3rd water-
planes, or between the 1st and the 4th waterplanes, the fall in the centre of buoy-
ancy, due to the rising of the vessel to any of these planes, may be determined.
We have here a means of constructing a diagram which will show the variation
in the height of the centre of buoyancy with change in the displacement, and
from which, therefore, the position of the centre of buoyancy for any draught may
be read off. Take a vertical scale of draughts A B (fig. 34), and spot off on it

Fig. 34.

the positions of the various centres of buoyancy as calculated for the vessel
when immersed to the 1st, 2nd, 3rd, etc., waterplanes.

Through these points, indicated by 6 2) b 3) etc., in the figure, set out
horizontally distances, b 2 h 2 , b 3 /? 3 , etc., equal to those between the load water-
plane and the waterplane to which each centre refers. A fair curve through
the points 6„ /z 2 , h :} , etc., will be the locus of centres of buoyancy required.

If now the height of the centre of buoyancy at any draught be required
it is only necessary to draw a line on the diagram parallel to the middle line
A B, and at a distance from it equal to that between the load-line and the
given draught; the point of intersection of this line with the locus gives the
required height of centre of buoyancy.

As showing the work in an actual case, let us construct the diagram for
the vessel whose centre of buoyancy at the load draught has already been

CENTRE OF BUOYANCY.

determined. Reverting to fig. 33, it will be necessary to find the areas PEQ
and DEPN.

The latter area may be obtained by the Five-Eight Rule already described ;
area PEQ, however, cannot be correctly found by this Rule. In this case we
should proceed as follows: — Multiply the near end ordinate, or A 19 by 3, the
middle ordinate, or A& by io, the far end ordinate, or A 3 , by - 1, and the
sum of these products by one twenty-fourth the square of the common inter-
val between the ordinates.

Arranged in tabular form, the figures of the calculation are : —

VOLUME.

MOMENT.

No. of
Ordinates.

Areas ot
W. Planes.

S\M.

Functions.

Areas of

w.rLuiL's.

Multiplier.

Functions.

80OO
7600
7000

- 1

40OOO
60800

- 7000

8000
7600
7000

10
- 1

24000

760OO

- 7000

93S00

930OO

Moment of layer D E P N
Volume of layer D E PN

93000 x 9
24

93 8o ° x 3
12

34S75

Distance of centre of buoyancy ^

of layer below No. 1 W.P., / ~~ 23450

= 34S75

= 23450 cubic feet.

= 1 "48 feet.

Having now got the position of the centre of buoyancy of the layer, the
distance the centre of buoyancy will fall when the vessel rises to the 2nd
waterplane may be easily determined, since the moment of the layer and the
moment of the displacement below the 2nd waterplane aLout the centre of
buoyancy of the total displacement are equal. The total volume of displace-
ment we have found to be 86800 cubic feet. The volume of the layer between
the 1st and 2nd waterplanes from our calculation is 23450 cubic feet; the
volume below the 2nd waterplane will therefore be, 86800 - 23450 = 63350
cubic feet. Calling d x the fall of centre of buoyancy in feet, we may write : —
d\ x 63350 = 23450(6 - 1-48)

23450 x 4-52
■ •*-' 63350 =l67

For the fall of the centre of buoyancy when the vessel is at the 3rd water-

3§

SHIP CONSTRUCTION AND CALCULATIONS.

plane, we must find the areas FEG and DEFH, which may be done thus.
using Simpson's First Rule ; —

No. of
W.P.

S'.M.

Areas.

Function of
Areas.

Moments from
Diayiam.*

FuiH'ti »n of
Jlrmieiila.

4
i

Sooo
71100
7000

SOOO

30400

7000

7600
I4000

30400
I4OOO

45400

44400

Distance of centre of buoyancy \
of layer below No. 1 W.P., /

44400 x - x 3*

45400 X

= 2*93 feet.

The volume of the layer = 45400 cubic feet, so that the volume below the
3rd waterplane will be, 86S00 - 45400 = 41400 cubic feet. If d 2 be the fall
of the centre of buoyancy in feet, we have —

d 2 x 41400 = 45400 (6 - 2'93)
45400 x 3-07

d, =

41400

3'3 6 -

d :i) the fall of the centre of buoyancy when the vessel floats at the 4th
waterplane, will be found to be 5 '10 feet. To find the requisite areas in the
diagram (fig. ^t,) the Three-Eight Rule should be used; otherwise, the calculation
is similar to the preceding ones.

dt> the fall of the centre of buoyancy to the 5th waterplane, is 6'8i feet
We have now sufficient information to construct the locus which, when plotted
as described, will be found to give the curve shown in fig. 34.

We now proceed to show how the longitudinal position of the centre of
buoyancy may be obtained. Reverting to fig. 32, the transverse sections
marked 1, il, 2, 3, etc., are those at which the elementary layers or slices
required for the moment calculation are taken, the number of divisions bein°-
arranged to suit the application of Simpson's First Rule. At each end inter-
mediate sections are introduced, the common distance being there reduced by
a half.

Having calculated the various vertical areas, we first take a horizontal line
A B equal on a convenient scale to the length of the vessel (fig. 35), and draw
lines at right angles to it at points in the length marked off to correspond with
the positions of the sections. On the upper side of A B we then plot the
sectional areas just found and draw a curve A C B through the extremities of

* Multiplication by the common interval is done at the end as shown.

CENTRE OF BUOYANCY.

the ordinntes, thus enclosing an area which represents the volume of the vessel
below No. i waterplane. On the lower side of A B, the moments of the layers
are set off. Station 6 is chosen as the axis of moments, as the areas of the
moments curves may then be obtained by Simpson's first rule. Calling the
vertical areas A 19 A^ A 2 , A 3 , etc., and the distance between them h, the moment
of an elementary layer of very small thickness a at section 6 will be zero; at
section 5, A 5 hct; at section 4, A 4 2ha; and so on to the left of the axis. To
the right of the axis we have at section 7 a moment A n ha\ at section 8,
A s 2ha; section 9, >4 3/7 a, etc. As in the previous case, the moments are
plotted as rectangles, the base a being, in each case, measured along the
axis, and the other side of the rectangle, represented by the area and lever
multiple appropriate to the section under consideration, erected as an ordinate.
Fair curves ADE and EFB are drawn through the extremities of these little
rectangles on each side of the axis as shown in fig. 35. The area enclosed by
each of these curves and the axis A B represents the longitudinal moment of
the volume on the side of the axis to which it refers.

Fig.

35.

«\S

H-

-C

(M.

-t

>->£-

< <

J->

t~*

By the principle of moments, we are obviously now able to write: —
Horizontal distance of centre of buoyancy) _ area ADE - area EFB
from axis through station No. 6 / area A G B

To illustrate the foregoing and show its application, take the following
numerical example : —

The areas in square feet of the vertical transverse sections of a vessel up
to the load waterplane are respectively, o, 20, 6o, 160, 230, 400, 750, 470, 350,
270, 100, 30, and o. The sections are 12 feet apart, except at each end,
where an intermediate one is introduced. It is required to determine the
longitudinal position of the centre of buoyancy.

From inspection, we note at once that section No. 6 will form a suitable
axis about which to take moments. Keeping this in view, and also remember-

SHIP CONSTRUCTION AND CALCULATIONS.

ing that, in calculating the moments, multiplication by the common interval is
left to the end, we are able to tabulate the figures as follows : —

No. of
Section.

Area of
Section.

S.M.

Function
of Areas.

Multiple

for
Leverage.

Function of
Moments.

rSo

I.l

360

[6 j

640

1920

230

460

920

5
6

400

75°
470

1600
1500

1880

1
1

1600

4980

1880

35°

700

1400

270

I080

3240

100

1. -1

I50

600

ioi

3°

270

1 1

—

8200

739°

ty <. f , r u (739° - 49 So ) — x 12
Distance of centre of buoyancy 3

from axis through section No. 6 ~ Q 12

0200 x —

3-52 ft. towards No. 7 section

Fig 86. &

\lA

\c
la

|J*

C '-

mid;

'hips

SCALE rOR LOCUS

We saw how to construct a diagram giving the vertical position of the
centre of buoyancy at all draughts; a diagram giving similar information can

CENTRE OF BUOYANCY. 4 1

be made for the longitudinal position. The work, however, will be more
laborious than in the previous case, as a new set of vertical areas must be
found corresponding to each draught, and a complete moment calculation,
similar to the one just worked out, made in each case. Assuming the horizon-
tal positions of the centre of buoyancy found up to a series of draughts
between the top of keel and the load-line, the diagram may be easily con-
structed. A vertical scale of draughts is taken, the horizontal distances of
the centre of buoyancy, as calculated from some chosen axis, are marked off
at the corresponding draughts, and a fair curve drawn through these points.
Fig. 36 illustrates this diagram.

QUESTIONS ON CHAPTER IB.

T. Tf two unequal weights be suspended one at either end of a weightless lever, find the
point at which the lever must be supported in order to be exactly balanced. If the lever be
48 inches long and the weights 11 lbs. and 5 lbs. respectively, find the balancing point from
the end loaded with 5 lbs.

Arts. — 33 inches.

2. If weights of 5, 8, 11, 13 and 17 lbs. lie on n table of rectangular outline, their posi-
tions taken in the order given and measuring Iioni one end being 3, 4, 5-5, 6, 7 "5 feet, and
from one side 1, 175, 2*5, 3, 275 feet, find the position of the point through which the re-
sultant force acts.

Ans. — 579 feet from end, 2*45 feet from side.

3. Define Centre of Gravity. — The equidistant \ ordinates of a vessel's watcrplanc,
beginning aft, are '2, 64, 10/2, 1 r 'O, 104, 7'o and '4 teet, and half ordinates introduced at
the extremities in the usual way have values 4 feet and 4*3 feet; find the distance of the
centre of gravity from the middle line.

Ans.—- 4*5S feet.

4. If the longitudinal distance between the ordinates in the preceding question be 14 feet,
calculate the position of the centre of gravity with reference to the No. 4 ordinate.

Ans. — *43 feet forward of No. 4 ordinate.

5. Define Centre of Buoyancy. — The area of a ship's load waterplane is 7000 squaie
feet, and the areas of other parallel waterplanes spaced 3 feet apart are respectively, 65CO.
5500, 4000, and 2000 square feet (neglecting the volume below the lowest section); obtain the
distance of the centre of buoyancy below the load waterplane.

Ans — 5*03 feet below load waterplane.

6. Referring to the previous question, calculate the fall in the position of the centre 0.1
buoyancy as the vessel rises to each of the given waterplanes, except the last, and plot the
locus of centres of buoyancy.

7. Explain how you would proceed to calculate the longitudinal position of the centre of
buoyancy? As a practical example, obtain the position of the longitudinal position of the centre
of buoyancy of a vessel, the areas of whose transverse vertical sections arc, starting from aft, 4,
100, 180, 240, 260 242, 190, 120, and 8 square feet, the sections being spaced 15 feet apait.

Ans. — I'ji feet forward of Xo. s section.

CHAPTER III.

Outlines of Construction.

AT this stage it is desirable to obtain an acquaintance, in a general way at
least, with the system of construction of the modern ship, and with

the names of the principal parts ; in a later chapter we shall take up
details.

In the old days, when wood was the medium of construction, vessels were
invariably built on what is known as the transverse system; and, as was perhaps
natural, the earliest iron ships, when that material began to displace wood,
were built on the same plan. There were, of course, important differences in
the details of construction due to the great difference in the nature of the
materials, but the general principle was in each case the same. As a founda-
tion and sort of backbone to the structure, there was, for instance, the keel
running fore-and-aft, and, at equal distances along the keel, transverse vertical
frames, or ribs, erected to give the form of the vessel at each point, and to
offer a convenient means of fitting the watertight skin or shell At their
upper end the transverse frames were joined by horizontal girders or beams,
adapted to keep the frames to their proper shape, and to support a horizontal
platform or deck. If the vessel were a large one, usually one or more decks
might be fitted below the upper one, this being necessary for strength, and it
might be for the convenience of stowing certain cargoes, or of housing
passengers.

In fig. 37, which is the midship section of a small steel vessel built on
the transverse system, are seen all the characteristics just referred to. A\ the
keel, is a steel or iron bar of considerable depth and thickness. The trans-
verse frames, marked F, are angles running from the keel to the gunwale, and
associated with bars of similar shape, called reverse frames from the circum-
stance of their looking in an opposite direction to the frames as shown in
horizontal section at A. At the bottom of the vessel, deep vertical plates
called floor plates, are riveted to the frames and the reverse frames, the reverse
frames being carried along at the upper edge of the floor plates, as shown.
BB are the beams, which as above mentioned, tie the sides of the ship and

OUTLINES OF CONSTRUCTION.

resist any tendency to change ot transverse form. Vertical change in
the transverse form is resisted by means of the pillars, which tie the
top and bottom parts of the structure together, and assist the floor plates
to carry the cargo. The longitudinal shape is maintained by means of the
keelsons and side stringers, which tie the transverse parts together, distribute
the stresses, and make the framing into one united structure. The most

Fig. 37.

S.S. 160-0 * 22-0x12-1

Lloyds Numerals

Transverse N* 34-92

Longitudinal W 5587

d = 11-87 FT V= 12-38

*SH£ER$TRAKE
v3r SSX-46T0-M

J6*-J6T0-3t

36T0 12

*-o TO -'HI

56 to 32

-36 to "32

K.7XI*

important part of all is the outer plating or skin, which gives the vessel its
floating power. It will be seen to consist of strakes of plating riveted to
each other and to the fore-and-aft flanges of the frames. The top deck is
also covered in with plating or wood to give strength, and keep the water out.
The strakes of plating running fore-and-aft on the outer ends of the beams, and

44 SHIP CONSTRUCTION AND CALCULATIONS.

connected to the shell by means of angles, are called stringers; they nre valu-
able elements of strength, as we shall see when we come to consider the
stresses to which a ship is liable.

From the diagram it will be seen that the material forming the structure
is not evenly distributed throughout. For instance, the shell-plating is thickest
at the top and at the bottom. Between the sheer strake, which runs in way
of the gunwale or the top deck at side, and the bilge strakes, the plating is
reduced in thickness, being a minimum about midway between these points.
Also the centre keelson is much heavier than the keelsons and stringers higher
up on the sides. The scantlings, too, are not the same right forward and aft
The 'midship thickness and sizes are only maintained for half the vessel's
length, or thereabouts, and then a gradually tapering process is begun, minimum
sizes being reached at the bow and stern. It should be mentioned that local
requirements usually demand heavier materials just at the extreme ends, but in
general the principle of reducing the scantlings as above is followed. We shall
see presently the reason for all this.

Fig. 37 illustrates only the simplest lorm of construction of steel vessels.
Departures have been made at different times, called forth by the desire of
the owners to increase the value of their property as producers of wealth, and
these departures have eventually resulted in considerable modifications in the
structure of vessels. Thus we have water-ballast tanks. When first introduced
these tanks were mere additions to the ship's load, but they have now become
incorporated in the structure, and, as we shall see, have added immensely to
its strength and safety. In vessels of wood, to build in such tanks was im-
possible, but in those of steel it is the natural thing to do, as the mild steel
used in modern shipbuilding, owing to its nature, can be manipulated in such
a manner as to ensure continuity of strength and absolute watertighlness with
the ballast tank as part of the hull.

Other departures brought about by commercial considerations have resulted
in modifications of the framing; these we shall consider in detail when we
come to deal more particularly with actual types of the modern cargo steamer.
So far we have only attempted to obtain some familiarity with the various
parts of a vessel's hull in order to follow intelligently a discussion of the
stresses and strains to which ships aie liable, which we propose to take up in
the next chapter.

CHAPTER IV.

Bending Moments, Shearing: Forces, Stresses,
and Strains.

IN this chapter we propose to speak of the stresses and' strains to which ships
are liable, and as the same principles are involved in calculations of the
strength of ships and of simple beams, it will help us to begin with the
simplest cases and gradually lead up to those which are more difficult.

Take a beam A B (fig. 38), fixed at one end and loaded with a weight W
tons at the other, and consider the system of forces in operation at any

Fig. 38.

n----.

■ ---.I,

---j

section. Take one at x feet from the extreme end B. Neglecting the weight
of the beam itself, we have here acting : —

(1) A bending moment = W X foot tons, tending to bend the beam as
shown dotted.*

(2) A shearing force = W tons, tending to cause the portion of the beam
C B to move downwards relatively to the portion A 0.

In this simple case, the bending moment obviously is a minimum at B
and a maximum at A, since it varies directly with x. Also the shearing force is

* The deflection is shown much exaggerated for clearness,
45

4 6

SHIP CONSTRUCTION AND CALCUTATIONS.

the sama for all sections of the beam from B to A. To express this in a diagram,
take a line E F (fig. 39) to represent the length of the beam. At E set up
an ordinate E G, representing on some scale the maximum bending moment,
W x AB foot tons. Join G F ; EGF is the diagram of bending moments.
From it, by simple measurement, we can obtain the value of the bending
moment acting at any point of the length of the beam. For instance, the

Fig. 39.

bending moment at a section of the beam is given by the ordinate C x G 2 of
the diagram, G l F being marked off equal to G B.

For the diagram of shearing forces we have merely to construct a rect-
angle E L M F, on EF as base, the side EL representing to scale the force W.

If, instead of being concentrated at the outer end, the load be spread

Fig. 40.

evenly over the surface of the beam (fig. 40), at any section X feet from /?,
we shall have : —

(1) Bending moment = w X x

X' 1

foot tons, w being the load per

foot of length. The curve of bending moments will now take the form K R F
(fig. 39), and is obviously a parabola having its axis vertical.
(2) Shearing force =* w X tons.

BENDING MOMENTS AND SHEARING FORCES.

The shearing force will thus vary directly with X, will be zero at B and a
maximum at A, where it will equal the total load. The shearing force diagram
will be a triangle such as ELF, EL giving to scale the shearing force at A.

Consider now a beam supported at each end and loaded in the middle.
Fig. 41 illustrates the case. A B is the length of the beam, W the load in tons,
and P the re-action at each support. At any section x feet from the middle of
the beam, the weight of the beam itself being neglected, we have : —
Bending moment = P (A - x) foot tons.
Shearing force = P tons

Fig. 41.

f*-i

The bending moment increases directly as X diminishes. It is therefore
a maximum at 0, the middle of the beam, and zero at either end A B y
fig. 42 being the diagram. The tendency here is for the beam to become
curved convex side downwards, the ends rising relatively to the middle, and it
is convenient to describe the bending moment as negative, the diagram AGS
being drawn below the line to indicate this. Where a bending moment gives

rise to an opposite tendency, that is for the upper side of the beam to become
convex, as in the previous examples, and in the case of beams supported at
the middle and loaded at each end or uniformly, it is described as positive, and
the diagram is drawn above the line. With regard to the shearing forces, it
should be noted that at sections to the left of the middle, the tendency is
to cause the left-hand portion of the beam to move upwards relatively to the
right, and at sections to the right of the middle the reverse of this, 'the forces
acting being conveniently described as positive and negative respectively. This

4 8

SHIP CONSTRUCTION AND CALCULATIONS.

is expressed in the diagram by plotting the shearing forces for sections to
the left of o above the base line, and those for sections to the right of 0,
below that line. The diagram takes the form A G H K L B, as shown in fig. 42.

As a numerical example, let the length of the beam be 20 feet, and the
concentrated load at the middle of it, 12 tons. Neglecting the weight of the
beam, the re-actions at the supports will each be 6 tons. At a point, say 2
feet to the left of 0, the middle point of the beam, we have : —
Bending moment = 6 (10 - 2) = 48 foot tons,
and shearing force = + 6 tons.

If the diagram (fig. 42) had been constructed for this beam, the values
above of bending moment and shearing force corresponding to a section 2 feet
from the middle of the beam, could have been obtained by reading off the
ordinates at the corresponding point in the diagram.

If, instead of concentrated at the middle, the load be distributed equally
throughout the length of the beam (fig. 43), the diagram of shearing forces and
bending moments will be modified somewhat from that given above. Calling the
length of the beam 2 /, and the load per foot w tons, we have for the re-aclion

Fig. 43.

at either end of the beam, / w tons. At any section of the beam, say x feet
to the left of the middle point, there will be acting —

(1) A bending moment - / w {J - x) (/-*) {l-x) = — (P - X 2 ) foot- tons.

2 2

(2) A shearing force = Iw - (/ - x) W = w X tons.

In plotting the diagrams, we note from the equation above that the curve
of bending moments will be a parabola, that it will have zero values at each
end of the beam, since X will there be equal to either +/ or - /, and a
maximum value at the middle where X = 0. The shearing force diagram is
obviously a straight line, since the ordinates vary directly as x ; it will have
a zero value at 0, and maximum values at the supports. Calling, as before,
the shearing forces to the left positive and those to the right negative, these
maximum values are, respectively, +lw tons and -Iw tons. Fig. 44 illustrates
the diagram for a distributed load, and it should be compared with fig. 42
for a concentrated load. As an exercise it would be interesting to construct
a diagram, assuming a distributed load, in an actual case, sav that of the

BENDING MOMENTS AND SHEARING FORCES.

20-feet beam previously mentioned; but we leave the student to do this for
himself.

Diagrams of bending moments and shearing forces may be derived by a
graphic process, and where the loads are irregularly distributed, as, for instance,
in ship problems, this graphic process is the one most convenient to follow.
We propose, therefore, to describe it briefly.

Fig. 44.

For this purpose, let us consider again the beam fixed at one end and
loaded uniformly. By the proposed method we must start with a curve or
diagram of loads. The weight on the beam, including its own weight from
A to /?, being so much per foot of length, may be represented by the rect-
angle A BCD (fig. 45). This load is supported by the re-action of the wall

Fig. 45.

on the portion of the beam embedded in it, but we shall only consider the
external forces acting on the beam from the wall face outwards.

Now, we know that at any point x feet from the end of the beam (fig.
45) the shearing force = w X tons ; that is, the ordinate of the shearing force
diagram equals the area of the diagram of loads from the end of the beam up

SHIP CONSTRUCTION AND CALCULATIONS.

to the point under consideration, and, therefore, as already shown, the diagram
is a triangle.

Take now the bending moment. For any section of the beam, say at G t

X X

bending moment — w X x — foot tons, or = shearing force x — foot tons,

that is, equals the area of the shearing force diagram from B to G.

To construct the bending moment diagram it is therefore only necessary
to take certain points on the beam, to calculate the area of the shearing force
diagram from the end of beam to these points, and to plot the bending
moments thus derived on a convenient scale. BLAB (fig. 45), is the form
that such a diagram would take in the present case.

Fig. 46,

The construction is quite as simple for a beam loaded uniformly and sup>-
ported at the ends. Fig. 46 illustrates this case, A B being the beam drawn to
a convenient scale and DABC the diagram of loads upon it between the
points of support, including its own weight. In plotting the diagram of shear-
ing forces we begin at, say, the left-hand point of support, at which the shearing
force is positive and equal to half the load, that is, to half the area of DABC
Its value may be plotted as A L. From L the diagram of shearing force falls
in a straight line, the value of an ordinate at any section, x feet say, from 0, the
middle point of the beam, being the shearing force at A minus the load repre-
sented by the portion of the area of the rectangle DA BO from A to the section.
At the half load and the re-action at A are equal, and there is therefore no
shearing force. At sections to the right of the load exceeds the re-action at
A and the shearing force is negative, reaching a maximum value at B as
previously shown.

For the bending moment of a beam loaded as described at a section x feet
from middle, we have deduced the equation —

Bending moment = — (P - x") foot tons,

/ and W having the values previously given.

BENDING MOMENTS AND SHEARING FORCES. 5 1

This may be written —

Bending moment = (/ - X) I l»

which obviously expresses the area of the shearing force diagram from A to the
point considered. Thus, having obtained the shearing force diagram, to get the
curve of bending moment, it is only necessary to calculate the area of the
former from either end of the beam to various points in its length, to plot
these areas as ordinates and to draw a fair curve through their extremities.
AM B A is the bending moment diagram for this beam.

As the same principles apply, we are now in a position to consider the
case of a floating vessel. In the first place, take a vessel, say a steamer, in
the "light" condition, that is to say, completely built, and with all machinery
aboard and water in boilers, but without bunker coal, cargo, or consumable
stores; and assume her to be floating freely and at rest in still water.

A moment's consideration will make it clear that a tendency to longitudinal
straining, with which we are here dealing, must be principally caused by the

Fig. 47*

action of the vertical forces made up of the vertical components of the water
pressures acting upwards, and of the weight of all the particles in the mass of
the vessel acting downwards. In fig. 47, ILL is a diagram of loads for a
"light" vessel. We shall show in detail, presently, when we consider the im-
portant case of a ship among waves, how such diagrams are constructed; in
the meantime it is sufficient to note that fig. 47 shows weight in excess of
buoyancy at each end and amidships, and elsewhere, except at one point forward,
buoyancy in excess of weight. The excess of weight is obviously due to the
small volume and the great weight of the structure at the extremities, and to
the concentration of the machinery amidships, and the excess of buoyancy to
the empty holds.

In fig. 47, diagrams of shearing forces and bending moments are also
shown. The curve of shearing forces at any point in the length we know to
be the area of the curve of loads from either end up to that point, reckoning
the portions of area above the axis positive and those below negative. In the
present case, the curve takes the form SSSS. In the same way, ordinates of
the curve of bending moments are given by the area of the diagram of shearing

* The diagrams represented by figures 47 and 48 are taken from a paper read before the
North East Coast Institution of Engineers and Shipbuilders, by Mr. Bergstr6m in 1 889,

52 SHIP CONSTRUCTION AND CALCULATIONS.

forces from either end up to the points in the length at which they occur. We
thus obtain the cuive M M M, the ordinates ot which are seen to be a maximum
about the middle or tlie forward ana afLer holds, and a minimum at about
middle length. With a homogeneous cargo filling the holds, the case becomes
considerably modified. The curve 01 loads is now as shown at L L L (fig. 48).
It is seen to cross and recross the oase line at many points. In way of
the machinery the weight is slightly in excess, out is much more so in the
mainhold. In the forward and after holds, buoyancy is again predominant,

Fig. 48.

while weight is in excess at the extreme ends. The shearing force curve now
crosses the axis at three points in the length, while the curve of B M has two
maximum values, one forward and one aft, tending to strain the vessel in
opposite directions.

The bending moment and consequent straining effects on a vessel in still
water are, as a rule, inconsiderable compared with those sbe rrjus'i withstand
when among waves at sea. It is then that the ultimate strength of the structure
is called out, in some cases with disastrous results.

Fig. 49.

Let us try to conceive for a moment the position of a vessel when in a
seaway. If the waves be of regular form and speed, the vessel may, at a
given instant, be in one of several positions. She may be traversing the
waves in a line at right angles to the crests, or be rolling in the trough between
the waves : or she may occupy some intermediate position with her length at an
oblique angle to the crest lines. The bending moment will be different in
every position, and the hull should be designed strong enough for the worst
case,

BENDING MOMENTS AND SHEARING TOkCES.

Of the above conditions, the first one, in which the vessel is assumed at
right angles to line of wave crests, has been most frequently investigated. It
is the condition in which longitudinal straining is greatest, and may, therefore,
in this respect be considered to include the other two cases.

Fig. 50.

Taking the first condition, we note that it has two critical phases. One
of these is indicated in fig. 49, where the vessel, assumed in full sea trim with
cargo aboard, but with stores and bunker coal consumed as at the end of a
voyage, is shown poised instantaneously in an upright position on the crest of
a wave, the latter being at mid-length. The other phase is when the vessel,

Fig. 51

complete with bunker coal and stores as well as cargo — her worst condition in
this case — has a trough amidships and a crest at each end (see fig. 50).

As we shall see presently, when we construct the diagrams, the bending
moments are reversed in the two cases. The general straining tendency, with
the crest amidships, ordinarily is to cause the middle to rise relatively to the

Fig. 52.

ends, as shown in fig. 51, and with the hollow amidships, to cause the middle
to sink relatively to the ends, as in fig. 52. These strains are known as
hogging and sagging, respectively.

Diagrams of shearing forces and bending moments for a vessel situated as
indicated in figs. 49 and 50, are constructed on the assumption that the waves
are stationary, and that the problem may be treated as a purely statical one.'

SklP CONSTRUCTION AND CALCtJLAflON§.

No note is ordinarily taken of the fact that the quick passage of waves
past a vessel, particularly one of relatively fine ends, has a tendency to develop
an up and down motion in her, altering her virtual weight and buoyancy from
moment to moment, and consequently directly affecting the magnitude of the
bending moment, and, therefore, the strains brought upon her. It may be
mentioned that, where they have been specially allowed for, these vertical
oscillations have been found to reduce hogging and increase sagging strains.

In these diagrams, too, it is not usual to allow for the difference in the
water pressures in the waves as compared with those in still water, although
it is known that they are less in the wave crests and greater in the hollows
than at the corresponding depths in still water, due to the effect of the orbital
motion of the water particles in the waves, the general effect tending to a re-
duction of both hogging and sagging bending moments. Other points which
are ignored, or are found impracticable so far to deal with, are the effect of
longitudinal oscillations, that is, pitching and 'scending, and of rolling motions,
although clearly they may have considerable influence on the bending moments.

It is obvious, then, that diagrams as ordinarily constructed are only ap-
proximately true, and should be used merely as a means of comparison between
vessels. When so employed, they are most valuable as a guide in new designs
for determining the lines to be followed in making departures in construction.

Taking a vessel, then, in the condition exhibited in fig. 49, viz., with a
wavef crest amidships, we begin, as in the simpler case of the vessel in still
water, by constructing a curve of loads. Such a curve, we know, shows the
difference of the forces of weight and buoyancy at all points in the length,
and to obtain it we must first find the values of these forces. A curve of
forces of buoyancy is easily drawn. It is only necessary to calculate the buoy-
ancy per foot of length, at various cross sections, usually taken at equal distances
apart, and then in a diagram, whose base line represents the length of the

The

diagrams represented by figures 53, 54, and 5$ are from Mr. Bergstr6m's paper
referred to on page 51.

t In these calculations it is usual to assume the wave to. have a length equal to the
length of the ship, and a height of j- of its length.

CURVES OK WEIGHT AND feUOYANCY. 55

vessel, to mark off the results on some convenient scale as ordinates at cor-
responding points. Thus we obtain the curve ABO (fig. 53).

To draw the curve of weights is more difficult. There are various ways
of doing this leading to the same result. One of them is to deal first with
the hull material, by calculating the weight per frame space of that which is
continuous at chosen points throughout the length, and plotting the results on
the same scale as employed for the buoyant forces at corresponding points on
the diagram containing the curve of buoyancy ; and then on the curve obtained
by penning a batten through these points, super-imposing the irregular weights,
such as bulkheads, stern-post, propeller and rudder, engines and boilers, tunnel
and shafting, coal in bunkers, cargo in holds, etc.

The irregular weights are conveniently plotted as rectangles on bases
extending over a portion of the length in the diagram corresponding to that
occupied by them on the vessel. In the case of bulkheads, however, the

weight is sometimes spread over a frame space, and in the case of coal and of
cargo, the weight per foot of hold space is plotted. In the example chosen,
a homogeneous cargo of a density to completely fill the holds and bring the
vessel to her load-line has been assumed. This is usual in strength calculations,
as it would be obviously impracticable to exactly allow for a general cargo,
owing to the difficulty of obtaining the positions and weights of the various
portions of it. The complete weight curve or diagram is of a very irregular
form, as will be seen from the figure. It should be noted that the weight
curve must be equal in area to the curve of buoyant forces and have its centre
of gravity in the same vertical line, as these are the conditions of equilibrium.
If the area of the weight curve be greater or less than that of the curve of
buoyancy, the vessel will not float at the assumed water-line but at a deeper or
shallower draught, as the case may be ; and if their centres of gravity be in
different verticals a trimming moment will be in operation, showing the
assumed line to, be wrong also as to trim.

Having got the curves of weight and buoyancy to correspond as re-
quired, we note that the actual load on the vessel at any point is the

$6 SklP CONSTRUCTION AND CALCULATIONS'.

unbalanced force acting at the point; that is the difference between the two
curves. These differences, measured at numerous ordinates and plotted to the
same scale as the diagrams of weight and buoyancy, give a curve, or diagram
of loads, marked L L L in fig. 54. The conditions of equilibrium required in
this case are that the area of the portion of the diagram above the base
line shall equal the area below that line ; also that the centres of gravity of
the upper and lower areas shall be in the same vertical.

To construct the curve of shearing forces is a simple matter, since the
area of the curve of loads from the end to any point, is the value of the
shearing force at that point. In the same way, the curve of bending moment
is obtained by integrating the diagram of shearing forces. These two curves
in the case assumed — that of an ordinary well-deck cargo steamer — take the
forms SSS and M M M in fig. 54.

In constructing diagrams of loads, shearing forces, and bending moments,
for the other extreme condition in which the vessel is astride two consecutive
waves, with a hollow amidships (fig. 50), the main point of difference is in the

Fig. 55.

curve of buoyancy, which will now take the form B B B *(fig. 53). The curve
of weights will remain as before,* but the curve of loads will, of course, be
of a different form, the weights being in excess amidships and the supporting
forces in excess at each end (fig. 55), the tendency being, as already pointed
out, to develop sagging strains amidships.

RESISTANCE TO CHANGE OF FORM.— The foregoing shearing forces
and bending moments give rise to stresses in the materials, and to con-
sequent tendencies to change of form in the structure. It is, of course
important to prevent the stresses on the materials becoming sufficient to cause
rupture, and the tendencies to change of form from becoming actual permanent

deformation. We shall show presently that this may be done in three ways

first, by increasing the weight of materials; second, and preferably, by judicious
disposition of materials ; third, by design of structure.

To simplify our explanations, we shall take the case of an ordinary rect-
' A ship is reall y a huge beam or girder, and, consequently,

* Except that the bunker coal and stores, assumed consumed in previous case musi
now be allowed for.

angular beam

kfcStStANCE TO CHANGE Of fOkM.

what is true for the simple beam when under shearing stresses and bending
moments, is also true for the ship.

A BCD, fig. 56, is a rectangular beam, which we will assume to be of some
elastic material, such as will yield equally under tensional or compressive stresses.

Fig. 56.

Draw a horizontal line at mid height, and at mid length also draw two vertical
lines, ad, bd, at a little distance apart. Now, place this beam on supports
at each end, and load it at the middle and observe what happens (fig. 57).

The beam will be seen to sag in the middle ; also the top surface D C will
be found reduced in length, the bottom surface increased, while E F, the line
at mid height, though taking the curve of the beam, will have its length

Fig.

58.

c f"

ill!

a a

b '

unchanged. The lines ac and b d, which were drawn vertically on the side
of the beam, are now inclined to each other, although still straight. Let fig.
58 be an enlarged sketch of this portion of the beam. The original straight
beam is shown by dotted, and the beam as bent, by full lines.

s«

SHIP CONSTRUCTION AND CALCULATIONS

Now the stress due to the external bending moment has obviously in-
creased a 6 to a 1 b\ and reduced dc to d l G l . There is thus a compressive
stress on the upper part of the beam, and a tensional stress on the lower part.
Since the strains are reduced from the outside of the beam to zero at the
middle, e f being unchanged in length, the stresses must be correspond-
ingly reduced. Also, it is clear that the strain, and, therefore, the stress at
any point in the height of the beam, varies in direct proportion to the distance
of that point from the line ef; for example, the strain at b is double that at
a point midway between b and /. The surface of which ef is a portion of
the section, is known as the neutral surface. Let us now consider the equili-
brium of the beam as loaded in fig. 56 at any section such as ac. Taking
the portion of beam to the right, there is, as we have seen, an external bend-
ing moment due to W. 2 . No other external forces are acting, if we neglect the
weight of the beam itself, so that this moment must be counteracted by the
sum of the moments of the molecular forces of the portion of beam to the
left acting at the section. Besides these moments there is due to the load

Fig. 59.

a vertical shearing force in operation tending to move the right-hand portion
of the beam upwards relatively to the left. This shearing force is counter-
acted by the resistance of the fibres of material to shearing. We shall return
to this point again.

In fig. 59 we show enlarged end and side views of the section a C. HA
is a section of the neutral surface and is called the neutral axis at ac. At
N A there is no stress due to bending. Above and below this line the mole-
cular stresses push and pull the beam, as shown by the arrows. As the beam
does not move in the direction of its length, these horizontal stresses must
neutralise each other, that is —

Pulling stress + pushing stresses = (1).

We have seen that the stress at any point of a section varies directly
with its distance from the neutral axis : if, therefore, we know the stress at
any one point either above or below HA, we are able to write down equation
(1), because in materials such as steel or wrought iron the resistance to com-
pression and tension, within the elastic limits, is the same.

When we know the external bending moment and the area and form of
the section of the beam, we are able to find the internal stress at any point

resistance to Change; of- form. 59

of the section. Let us find it at unit distance, say one inch from the
neutral axis. Calling the stress in tons per square inch at this • point s,
at two inches from NA it will be 2 s tons, and, generally, at y inches either
above or below NA, it will be —

ys tons.

On a small portion ol area a, at this distance from the neutral axis,
the stress will be —

ysa tons
and the total stress acting at the section will be the sum of all such elements ;
we may therefore write : —

Total pushing and pulling stresses at section a C = S^yct tons,
where the symbol 2 signifies that the sum of the elementary forces is taken.
Now there must be no resultant stress acting at the section, so that —

s^yct = 0.
But ~yct is the moment of the area, and for this to be zero, the neutral
axis, about which the moments have been taken, must pass through the centre
of gravity of the area of the section. This fixes the neutral axis, and -is
an important point to remember. To get now the stress at a point one inch
from this axis as required, we must equate the sum of the moments of the
internal stresses about the neutral axis to M, the external bending moment
at the section.

At any distance y inches from NA, either above or below, the moment of
the stress acting on a small portion of area a is syaxy — sy 2 a inch tons.
And for the whole section we may write : — Sum of moments of internal
stresses = §2(/ 2 a inch tons.

Now, the expression 2*/ 2 a is a well-known quantity in physics : it is called

the moment of inertia of the section of the beam. If it be represented bv I,

and the internal and external moments be. equated, we get: — sI=M. So that

M
8, the stress in tons at one inch from NA,= -j--

To find the value of the stress at any point in the section, it is only

necessary to multiply s by the distance of the point from the neutral axis.

Thus, if y inches be the distance of the upper or lower surface of the beam

from NA, and p be the stress there, we shall have :—

• 1 % , s

Maximum compressive or tensional stress in tons at section a C =p=y -j-. (2)

This is the formula which must always be employed when dealing with
the strength of beams and girders, such as ships, and is worthy of careful
study. It shows that the maximum stress varies directly as M, the external

bending moment, and inversely as — • Consequently, with a given bending

moment, the stress is reduced by increasing this quantity and increased by
reducing it.

It is easy now to understand why in a ship or other loaded beam a
reduction of stress is effected by increasing the sectional area, that is, the

60 SHIP CONSTRUCTION AND CALCULATIONS.

weight of materials, by judiciously disposing them, or by changing the design.
It merely means that in each case the modulus — is increased. Increase of

sectional area directly increases the moment of inertia /; the same effect is
attained without increase of sectional area by concentrating the latter at points
remote from the neutral axis. Deepening the beam or girder will affect the
modulus in two ways ; / will be increased, but so also will y ; as / varies as
y% however, the effect on the whole will be to increase the modulus and reduce
the stress.

It is for these reasons that in the case of a ship it is desirable to
have the thickest plates at the upper deck stringers and sheerstrakes, and
at the keel and bottom plates, and the thinnest plates midway between
deck and bottom, the vicinity of the neutral axis in a ship. The above
formula, indeed, tells us that at the neutral axis there is no stress on the
materials due to bending moment, and it would thus appear that the scantlings
at the neutral axis might be reduced indefinitely. It happens, however, that at
this place in the depth a horizontal sliding or shearing action, which is de-
veloped by the variation of the benJing moment from point to point of the
length, and which tends to push the upper and lower portions of the structure
in opposite directions, has its maximum value. To counteract this straining
tendency, a considerable sectional area of material -is needed in the vicinity of
the neutral axis. We shall deal more fully with this point presently.

Another reason against thinning down too much the side plating of ships
is found in the consideration that when rolling excessively at sea, the sides
may frequently become, approximately at least, the top and bottom of the
girder, and be called upon to withstand considerable bending stresses.

To apply formula (2) to find the stress at any point of a beam, we must
know three things. We must know the external bending moment at the section
containing the point, the position of the centre of gravity of the area of the
section, and the value of the moment of inertia of the sectional area about
a horizontal axis through its centre of gravity. We propose to show in detail
how the work is done in the case of a ship, but before dealing with so com-
plex a girder, we shall take one or two practical examples of simple beams.
In a previous page we have explained how the external bending moment may
always be found ; we may therefore assume this item as known. Take then
as a first case, a steel beam of rectangular section 20 feet long, 1 2 inches
deep, and 3 inches thick, under a bending moment of 600 inch-tons at the
middle of its length, and let us determine the maximum stress on the material.
Begin by writing down the stress formula, viz. —

In the above beam the centre of gravity is at mid-depth ; therefore
y = 6 ins. Also the formula for the moment of inertia of the section in this

case a rectangle, about a horizontal axis through its centre of gravity, is ---,

RESISTANCE TO CHANGE OF FORM.

where A is the area of the section and h the full depth. Substituting the
given values, we have —

7 ^6X12X12 - .

I = ^ — — ^432 in. 4

1 2

. , r 600 x 6 . .

and therefore p = — 8'3 tons per sq. inch.

Fig. 60.

> C

1 *• 1

Taking the strength of steel at 30 tons per square inch, this stress allows
a factor of safety of rather more than 3-^-, which, in most cases, would be too
low, 5 to 6 being common for ship work. The form* of section above given
is by no means the most economical for steel beams. This material admits of
being rolled into many forms, and to show the great importance of distribution
of material as a means of increasing the strength of beams against bending, let
us assume the length, depth, and sectional area, and therefore the weight, to
remain as before, but the form to be as in fig. 60. The only additional work

* Within elastic limits mild steel and wrought iron are equally strong under compression or
tension, but this is by no means true of all materials. Cast iron, for instance, will withstand a
six times greater stress under compression than under tension : wood, on the other hand, has
its greatest strength under tension. In such cases, for maximum strength on minimum weight,

Fig. 61.

the section must be ol special design. The axis of moments must still pass through the
centre of gravity, but the stress may be reduced on the weaker side by concentrating the
material on that side near the neutral axis. For example, beams loaded at the middle and
supported at the ends, if of cast iron, to be of economical design should^ have cross sections,
pf such forms as indicated in fig. 61,

62 SHIP CONSTRUCTION AND CALCULATIONS.

to be done here is to find the moment of inertia of the new section about the
neutral axis, which, as in the previous case, is at mid-height. The formula for
the moment of inertia in this case is —

/= BH s -2bh s

where H is the full depth of beam, h the distance between the flanges, B
the full breadth, and b the breadth from the outer edge of the flange to the
side of the web. Substituting the values given in fig. 60 —

t- nx 1728-2 x6x 1000 n • 4

/= _•? i = 872 in.

We therefore have —

Stress at top and bottom of beam — — -— — — 4*1 tons per square inch,

872

a maximum stress which is only half of the previous one.

The above is only an illustration ; for various reasons, girders of this
section are not usually rolled with flanges of greater width than 6 to 7 inches.
Taking them at 7 inches, and increasing their thickness to if inches say,
with the same weight of material, a girder of 18 inches depth and i T 3 F inches
web could be obtained. The moment of inertia of such a girder would be

1685; and, under the same bending moment of 600 inch-tons, the stress on

the upper and lower flanges would be —

600 x 9
p = ~i68T~~ = 3' 2 tons P er square inch.

Let us turn now to the case of a floating ship. We have seen how to
obtain the external bending moment, and to apply the stress formula, it only
remains to determine for the material at the transverse section under the maxi-
mum bending moment, a method of fixing the position of the neutral axis, and
of calculating the moment of inertia about that axis. Now, as we know that
the neutral axis passes through the centre of gravity of the sectional area, its
position may therefore be easily found. As we shall see presently, the calcula-
tion involved is conducted simultaneously with that for the moment of inertia.

In setting out to find the moment of inertia we must bear in mind that
we are dealing with a built girder, and that only continuous material lying in a
longitudinal direction is to be considered as available for resisting longitudinal
strains. In ordinary cases the maximum bending moment occurs at about mid-
length ; we must therefore choose the weakest section in this vicinity for the
moment of inertia calculation, as, of course, if straining were to take place, it
would be at this section. Careful note should be made of the fact that
material under tension must be calculated minus the area of the holes for the
rivets joining the frames to the shell plating, the beams to the deck-plating,

RESISTANCE TO CHANGE OF FORM. 63

etc. This precaution need not be taken with the material in compression, as
the rivet, if well fitted, will be as effective to resist this stress as the unpunched
plate. Where continuous wood decks are fitted, they are sometimes allowed
for, wood being considered equivalent to about T \ of its sectional area in steel.
In the case of tension, this must be reduced on account of the butts, which
are usually separated by three passing strakes, also on account of the bolt
holes. For compression the full area is taken. In modern cargo steamers
continuous wood decks are seldom fitted, and there are none in the vessel
whose moment of inertia calculation is given below.

As already mentioned, the conditions dealt with in these strength calcu-
lations are those depicted in figs. 49 and 50. In the first case, hogging
strains usually predominate in ordinary vessels, the upper material being in
tension and the lower in compression. In the second case, sagging strains
would be expected, causing compressive stresses in the upper works and ten-
sional stresses below. Since the rivet holes require to be deducted from
the upper material in the moment of inertia calculations for hogging strains,
and from the lower material in that for sagging strains, obviously a separate
calculation is needed for each case. Dr. Bruhn* has pointed out that the
necessity of two calculations may be obviated by obtaining the moment of
inertia without correcting for the rivet holes, the stress so obtained being after-
wards increased inversely with the reduced sectional area. The results obtained
by this method do not differ much from those by the ordinary one, while
the work is less.

In the following example we show in detail how to find the moment
of inertia for a cargo steamer of modern type subjected to a hogging bending
moment. It will be observed that the full sectional areas are tabulated, the
sum of those of the parts in tension being reduced by \ as an allowance for
a line of rivet holes at a frame. It will also be noted that the moment
of inertia, in the first instance, is obtained about an assumed axis, the
position of the neutral axis being unknown ; that the distance between the
neutral axis and the assumed one is next determined, and that the value
of the moment of inertia about the neutral axis, which is what we require,
is obtained from that about the assumed axis, by employing the well-known
property of the moment of inertia expressed by the formula : I = I x - A h 2 .

Where / = moment of inertia about neutral axis.
I x = moment of inertia about assumed axis.
A = area of material in section.
h = distance between axes.

This principle is also employed in the first instance to obtain the moment
of inertia for each item about the assumed axis. For items of small scant-
lings in the direction of the depth of girder, the moment of inertia is expressed
with sufficient accuracy by multiplying the areas by the squares of their dis-

- See his paper on Stresses at the Discontinuities of a Shifs Structure^ read before the
Institution of Naval Architects in 1899,

6 4

SHIP CONSTRUCTION AND CALCULATIONS.

tances from the axis as in column 7. In the case of the side plating, however,
and the vertical parts of the double bottom, such as the centre girder, margin
plate, and intereostals, the figures of column 6 have to be increased by the
moment of inertia of each item about an axis through its centre of gravity,
that is, T V A d\ where d is the depth of the item, and A the sectional area ;
these quantities are given in column S.

The moment of inertia being obtained, the stress in tons per square inch
on material at any distance //, either above or below the neutral axis, is

quickly found, since p = — y.

It should be remarked that when applied to a large girder like a ship,
special units are employed, M being in foot-tons, A in sq. inches, y in feet,
/ in feet- and inches 2 .

Moment of Inertia Calculation.

(Ship under a hogging strain).

S.S. 350' o" x 50' 74" x 28' o". Assumed neutral axis above base, 16' o"
Depth from base line to bridge deck, 36' 10 feet.

Below Assumed Axis.

Items.

Scantlings in
Inches.

Sectional

Areas

= A

C.G. )
from > =h
N.A. J

Moments
of Areas

^ X /I

A x A*

a =

Depth of
Items.

\ Centre Girder,

44 * inr

1 IO

14*25

I5 6 '7

203

2233

Top Angle,

4M^jj

3*8

12-5

47*5

156

593

Bottom Angle,

4J x 4-i x -i §

S'o

i6'o

Scro

256

1280

In. Bot. Plating,

255xA

102 'O

i 2 '5

1275-0

156

15912

Margin Plate,

34 x^

i7'3

14-1

2 43'9

199

3443

Margin Angle,

4 x 4 X ^

3*4

15*4

5 2 '3

237

806

Side Stringer,

( 4 x -JS \

8-9

3*6

320

116

)»

8'9

8-5

75' 6

6|i

h Keel Plate,

21 x§£

21'0

16-1

338-i

2 59

5439

8 Strake,

54 x^

40'5

15*95

646*0

2 54

10287

G „

58 x JJ

37*7

15*35

597*5

25'

9463

D „

56 x -1 1

33"6

15*7

527*5

246

8266

E „

56x^§

364

15-6

567"S

243

8845

F „

53 x|J

34-8

i5"5

539"4

240

8352

G „

57 41

37 '0

i3'3

492-1

177

6 549

H „

57x^

34'2

9 -6

328-3

3146

J „

58x-i§

377

5'3

199-8

1056

K „ (Part),

4ox||

24*0

1-65

39*6

2 2

497'2

6239-1

86499
219

1
219

497*2

S6718

MOMENT OF INERTIA CALCULATION.

Above Assumed Axis.

Items.

Scantlings in
Inches.

Sectional
Areas

C.G. )
from } =h
N.A. )

Moments

of Areas

A xh

/j2

Ah*

& Ad*

d =

Depth of

Items.

Bridge Deck

Stringer,
Bridge Deck

Angle,
Bridge Deck

Plating,
Upper Deck

Stringer,
Upper Deck

Plating,

Side Stringer,
P Strake, -

„

H »

M „ - -

L „

K „ (Part),

42X-£§

4h x 4j x -ii

144 x ^

58xi§

i53xA
/6|x4ix^|

40x^2

49X-|^
44*-|£
55 *U
57x|$
17 x^

2 1*0

4'6

54*°
29*0

61*3

8-9

8'9
24*0
27*0
30-8
33 *o
37 'o

IO*2

19*2

^S
I9-8

12*3

12-85

6*6

i*4
17*9

14*7
11*2

7'5

3*3

403*2

88*i
1069*2

7877
587

12*5
429*6

39 6 '9
345'°
247*5
122*1

7"i

3^
367
392
151
165

320

216

125

7749

1688
21168

4379
10114

383

18
7680
5832
3850
1848
407

22
38
32
58

68
2

Less \ for rivet holes,

349*7
49*9

4324*3
617-7

65116
9302

220
3i

299*8

3706*6

558i4
189

189

Above assumed N.A.,
Below assumed N.A.,

299*8
497*2

3706*6
6239*1

56003
86718

797*0

2 532-5

[42721
2

285442

N.A. below assumed axis = _£3 — 5— 3*18 ft.

797

N.A. above base = 16-3*18 — 12-82 ft.

y — distance of top of vessel from N.A. = 36*0- 12*82 = 23*18 ft.
/_ 28544 2

23*18

12314.

The load displacement of the above vessel is 9600 tons, the draught being
23 ft. 9 ins. ; if we assume, as is frequently done in approximate calculations,
that the maximum bending moment on the wave crest is equal to a thirty-fifth

66 SHIP CONSTRUCTION AND CALCULATIONS.

of the displacement multiplied by the length, we shall have in the present

instance : —

Maximum bending moment = § 5l_ = 96000 ft. tons:

and if we use this figure with that just obtained for the value of — , we shall

get for the greatest stress acting on the vessel when under a hogging strain —
M ,

— r QOOOO . .

pe / = — = 779 tons per square inch.

~y I2314

To obtain the greatest stress under a sagging strain, as previously pointed
out, a new moment of inertia calculation is necessary, otherwise the work is
similar to that just explained and need not be here detailed.

With regard to the magnitudes of calculated stresses, it may be said that,
generally speaking, these increase with size of vessel. Small vessels have to be
built to resist local strains, and are probably too strong, considered as floating
girders. At anyrate, their actual calculated stresses, of which records are avail-
able, show these to be very small indeed. In 1874, Mr. John investigated the
longitudinal strength of iron vessels of from 100 to 3000 gross tonnage, on the
basis of Lloyd's scantlings, the following being some of his results : —

Gross tonnage of Ship. Tensional siress in tons per square inch

at Upper Deck.
IOO 1*67

5 00 3"95

IOOO 5'2

2000 5*9

3000 8*09

Later calculations for steel vessels of large size, which have proved satisfactory as
to strength, show maximum stresses of between 8 and 9 tons and even higher
The Servia, a passenger and cargo vessel of 515 ft., had a calculated stress at the
upper deck of 10*2 tons per square inch when on the wave crest, and of 8 - o4
tons when in the wave hollow, while the Maurelam'a*, of 760 ft. length, is stated
to have a calculated maximum stress of 10*3 tons on the top member. It should
be added that a special high tensile steel was largely used in the construction
of the upper works of the latter vessel.

With regard to compressive stresses, it is important to note that thin deck
plating is liable to buckle when in severe compression, and is therefore not so
efficient under a sagging as under a hogging strain ; this should be borne in mind,
particularly when considering maximum stresses on bridge and awning-decks.

It has already been pointed out that stresses, such as the above, are
not the actual stresses experienced by the vessel, since the conditions of figs.
49 and 50 do not fully represent those of a vessel among waves. The results,
however, are valuable for comparison. In the case of a proposed vessel, for
example, if the calculated stress be not greater than in existing vessels whose

* See the Shipbuilder for November, 1907.

SHEARING STRESSES.

recoids have been satisfactory, the scantling arrangements in the new ship may
be considered adequate. If it be much greater, so as to approximate to the
calculated stress in vessels which have shown manifest signs of weakness
when on service, then additional strength must be added. From our previous
considerations, it will be clear that the most economical position for the
new material to resist bending, will be either at the top or bottom of the
vessel, /.*., as far as possible from the neutral axis, as it will there be of
maximum efficiency in reducing the stress.

SHEARING STRESSES.— We come now to consider the effect of shear-
ing forces on a structure. We have already explained how the values of such
^forces may be obtained at all points in the lengths of beams, including floating
vessels, under various systems of loading, and we have now to determine the
stresses caused thereby.

Fig. 62.

If the vertical shearing force at any section be taken as F, we may
obviously write : —

Mean stress per square inch / _ F_
due to shearing force \ A '

where A is the number of square inches of material in the section. For
example, if a rectangular beam of section 8 inches by 4 inches be under a
shearing force of 64 tons, then —

Mean stress per square inch =

= 2 tons.

8x4

The actual stress at any point of the section may, however, be very different
from this mean stress, as we shall now proceed to show.

In fig. 62 we have the diagram of bending moments for a beam sup-
ported at the middle and loaded at each end. The bending moment is a
maximum at the point of application of the support, and has zero values at

SHIP CONSTRUCTION AND CALCULATIONS.

each end. For any two vertical sections A x A 2 and A 3 A^ the bending moments
may be read from the diagram. Section Aj A 2 being nearer mid-length has
the greater bending moment. /!//!/, the neutral surface, may be considered to
divide the beam into two portions, of which the upper one is in tension and
the lower in compression.

Consider now the equilibrium of a small portion of the beam A X RLA Z ,
shown enlarged in fig. 63. There are pulling forces acting on the end A r R
and on the end A 3 L. As the former section is under a greater bending
moment than the latter, the stresses will also be greater. There" will' thus
be a force equal to the difference of the total forces acting on the ends
tending to move the portion of the beam A 1 R L A z towards mid-length.
This action is balanced by a shearing force over the bottom surface - L R.
Clearly, the magnitude of this shearing force will vary with the areas of the
ends A X R and A 3 L At the top of the beam the shearing force will be
zero, and will gradually increase as R L approaches N /I/, where it will be a

Fig. 63.

maximum. Below the neutral surface, the forces act in opposite directions,
and therefore as RL approaches the lower end of the beam, the shearing force
will gradually be reduced, becoming again zero at An A 4 .

If the vertical sections be at; unit distance apart, say one inch, the horizon-
tal shearing stress pei square inch at any point- Z. - of' -section A 3 A 4 will be
obtained by dividing the shearing force acting on the surface R L by the
breadth in inches of the beam at that point. This is also the value .of the
vertical shearing stress on the section at the same point, since there cannot be
a shearing stress in one plane of a beam without an equal one at the same
point in a plane perpendicular to the first. Proceeding in this way, we arrive
at the following formula for the shearing stress per square inch at any point of
a cross section : —

where A — Area in square inches of the portion of the cross section above or

below the given point.
g = Distance in inches of the centre of gravity of the area from the

neutral surface.
F = Vertical shearing force at the cross section in tons.

SHEARING STRESSES. 69

/ = Moment of inertia of the whole section (in inches 4 ).

q = Stress per square inch at the given point.

b = Breadth of beam in inches at the given point.
Let us apply the formula to the case of the rectangular beam whose mean
shear stress was found above to be 2 tons. Substituting values, we get for the
stress intensity at the neutral axis : —

16 X 2 x 12 x 64 , -1

a = 3 = 3 tons per square inch.

4 x 32 x 64

Thus the maximum shear stress is, in this instance, 50 per cent, greater than
the mean.

The above is for a simple beam of rectangular section, but the same
formula may also be applied to the more complex case of a ship. In the
latter instance, of course, the beam is of hollow section, and b will be twice
the thickness of the shell plating. It is important to note that only continu-
ous longitudinal materials must be used in rinding A. Obviously, the value
of the shearing stress will vary with P, the vertical shearing force, which is a
maximum in ordinary vessels at about one-fourth the vessel's length from each
end ; so that at the neutral axis at these points of the length the shearing
stress may be considerable. We see now why it is inadvisable to unduly
reduce the scantlings in the vicinity of the neutral axis.

It is also important to give special attention to the rivets in the landings,
or longitudinal seams, in this neighbourhood, as the shear stress gives rise
to a tendency for the edge of one strake to slide over that of the next.
Recent experience with large cargo vessels has shown that the usual plan of
double riveting the seams is only sufficient for vessels up to a certain size,
say 450 or 480 feet in, length. Longer vessels will develop weakness at the
longitudinal seams unless precautions be taken to increase the strength of the
riveting. Lloyd's Rules now require treble riveted edge seams in the neigh-
bourhood of the neutral axis in the fore and after bodies in vessels of the
above length and beyond.

TRANSVERSE STRAINS.— So far, we have dealt exclusively with stresses
which tend to strain a vessel longitudinally, and while such stresses are prob-
ably of first importance, we must not omit to refer to those which come upon
a vessel in other directions.

It has been customary to consider stresses which tend to change the trans-
verse form as next in importance to those affecting a vessel longitudinally.
Structural stresses in other directions are, indeed, partly, longitudinal and partly
transverse, and where the predominant stresses are known for any vessel, the
effect of their combination in a diagonal direction may be predicted. Unfortu-
nately, the subject of transverse stresses of ships is a complicated one, and we
cannot do more here than indicate generally the external forces which operate
on a vessel so as to alter her transverse form, and point out the structural
arrangements which best resist this deforming tendency.

7°

SHIP CONSTRUCTION AND CALCULATIONS.

Consider in this connection the case of a vessel afloat in still water (fig. 64).
The hull surface is pressed everywhere at right angles by the water pressures,
indicated in the figure by arrows, and the resulting tendency is towards a general
deformation of the vessel's form. Taking the transverse components of the
water pressures, these obviously tend to force up the bottom and press in the
sides, as shown exaggerated in fig. 64. Such tendencies, however, are pre-
vented from becoming actual strains by the internal framing. The compara-
tively thin shell plating, which might yield under heavy water pressure, particularly
in the way of an empty compartment, is kept in shape by the frames, rigidly
connected to the beams and to the floors at their top and bottom ends re-
spectively, and supported between these points by hold stringers and keelsons.
In way of the bottom, the deep floors, spaced at comparatively short intervals,
and fitted, in the first instance, as supports to the cargo, are splendid preservers
of the form. The floors, too, are tied to the beams of the decks by means of

Fig. 64.

TTTrrr^

strong pillars, and in this way a stress which comes upon one part of the struc-
ture is communicated to it as a whole. Probably the most efficient preservers
of transverse form are the athwartship steel bulkheads. Where these occur
the vessel may be considered as absolutely rigid, and care should be taken
to spread this excess of strength over the space unsupported by bulkheads
by means of keelsons and hold stringers.

Docking Stresses. — A vessel when docked or when aground on the keel
particularly if loaded, has to withstand severe transverse tresses. The re-
action of the weight at the middle line will tend to force up her bottom
while the weight of cargo out in the wings will set up a considerable transverse
bending moment and cause the bilges to have a drooping tendency. This is
shown, much exaggerated of course, in fig. 65. There will be tensile stresses
of considerable magnitude acting along the top edges of the floors ; and if the
vessel be one having ordinary floors, weakness may be developed at the lower
turn of the bilge, as the framing has there to withstand a shearing stress due
to the weight of the cargo above. The floors should therefore be kept as deep

transverse strains.

as possible at the bilge, and should be carried well up the sides. In vessels
having double bottoms this part of the structure is very strong owing to the
deep wing brackets, which bring the resisting powers of the side framing into
operation.

The straining at the middle line will be arrested by the pillars, if efficiently

Fig. 65.

fitted; these will act as struts and communicate the stresses to the deck beams,
which will resist a tendency to spring in the middle and to bring the sides
together. Thus, as in the case of still water tresses, the straining tendency
will be resisted by the structure as a whole. This interdependence of parts,
causing equal distribution of stress throughout, is what should be aimed at in
design, and special pains should be taken to ensure efficient connections.

Fig. 66.

Transverse Stresses due to Incorrect Loading, — A preventable cause
of transverse straining is that due to the manner in which heavy deadweight
cargoes are sometimes loaded. Frequently, the heaviest items are secured at
the middle line of the vessel instead of being spread over the bottom, the
wings having therefore comparatively little weight to carry. The straining ten-

SHIP CONSTRUCTION AND CALCULATIONS.

dency in such a case is to elongate the transverse form, the water pressures on
the sides tending to the same end. This condition is illustrated in fig. 66, the
dotted lines representing the normal vessel, and the full lines the vessel as
strained. The pillars will be here called upon to tie the top and bottom of
the structure, but not unfrequently the rivets connecting the pillars at top and
bottom have been sheared in places with consequent dropping of the bottom
part of the hull.

Transverse Stresses due to Rolling. — We have pointed out that it is
when among waves at sea a vessel meets with the most trying longitudinal
stresses, and it may now be added that tendencies to transverse straining are
also greatest then. These latter stresses probably reach maximum values when
a vessel is rolling in a beam sea, and they are obviously due to the resistance
which the mass of the structure offers to change of the direction of motion

each time the vessel completes an oscillation in one direction and is about to
return. The stress is a racking one, and tends to alter the angle between the

Fig. 67.

deck and the sides, also to close the bilge on one side and to open it on the
other. Such a racking strain is exhibited graphically in fig. 67.

The parts of the structure most effective in preventing this change of
form are the beam knees, transverse bulkheads, web frames or partial bulk-
heads, and the ordinary side frames, in which is included the reverse bar, if
any. The beam knees should be of good size, efficiently connected to the
frames and beams, and fitted well into the corner formed by the side plating
and the deck. Change of form at the bulkheads is practically impossible, if
they be stiffened sufficiently against collapsing; careful attention should there-
fore be given to this point. The side frames, owing to their position and
close spacing, offer powerful resistance to racking, but in order to attain maxi-
mum efficiency they should be securely riveted to the beam knees, and the
floors or tank brackets should be carried well up the sides. These brackets
virtually reduce the length of the frame, and it is well known that reducing
the length in such a case greatly increases the rigidity

TRANSVERSE STRAINS. 73

Local Stresses. — Besides longitudinal and transverse structural stresses,
vessels have to resist other straining tendencies due to local causes. For
example, the engines and boilers with their seatings together form a heavy
permanent load on a comparatively small fraction of the length, and thus give
rise to considerable local stresses. These are provided against in various ways,
some details of which are given in a later chapter. It may be said that the
general principle is to increase the strength of the structure in way of the
loaded zone, and, by means of longitudinal girders and otherwise, distribute
the load to the less strained portions of the hull beyond.

Other stresses due to the propelling machinery are those brought on the
stern of the vessel by the action of the propeller itself. These are most severe
when the vessel is rolling and pitching among the waves, and consist chiefly
of vibrations caused by the frequent racing of the propeller and checking of
the same, as it rises out of and sinks into the water. The parts that suffer
most are the connections of the stern frame to the vessel, and it is highly im-
portant, therefore, that these should be made amply strong. We shall see
presently, when we come to consider details of construction, what the usual
arrangements are in such cases.

Panting Strains. — These strains, which are usually developed in the shell
plating forward and aft, where it is comparatively flat, consist of pulsating
movements of the plating, as the name indicates. They are partly due to blows
from the sea, and partly to the resistance offered by the water to the vessel's
progress as she is driven forward by the propeller. An ordinary cargo vessel
is not so much troubled by these strains as a fine-lined passenger steamer,
for she is slower, and her full ends are better able to resist a tendency to
flexibility than the flatter form of the faster boat.

The usual means taken to strengthen the shell against panting, is to fit a
hold stringer in the vicinity affected, and connect it well to the shell plating
and framing ; if the vessel be fairly large, the stringer should be associated
with a short tier of beams, which act as struts and prevent movement in the
plating. In very fine vessels the floor-plates should be deepened forward and
aft. A panting arrangement for a cargo steamer is shown in the chapter on
practical details.

A class of strains somewhat akin to those of panting are frequently found
developed under the bows of full cargo vessels, in the shape of loose rivets
and generally shattered riveted connections. They are now recognised to be
due to the pounding which a vessel receives from the waves as she rises and
falls among them. As might be expected, they are found much aggravated
after a voyage made in ballast trim, for the pitching motions will one instant
lift the fore end high out of the water, and the next bring it into it with terrific
force. It is scarcely wonderful that this pounding, repeated throughout a
long voyage, should produce the results mentioned. Obviously, efficient ballast-
ing is of vital importance, and that this is the opinion of those having shipping
interests, was evidenced by the appointment of the Royal Commission, under
Lord Muskerry, to consider the desirability of fixing a minimum load-line to

74 SHIP CONSTRUCTION AND CALCULATIONS.

sea-going vessels, although, for various reasons, there was no practical result
therefrom.

Strains due to Deck Loads, etc. — These loads consist of steam winches,
the windlass, donkey boilers, steering gear, etc. The resulting stresses can
usually be counteracted by an efficient system of pillaring, with perhaps a few
extra beams if the weights be very great.

The Racking Strains brought on the deck of a sailing vessel by stresses
from the rigging and masts should be mentioned. In sailing vessels not of
sufficient size to require a steel deck, special tie-plates should be arranged in
a diagonal direction so as to communicate the stresses from the plating round
the mast to the deck beams and side stringers, to all of which the tie-plates
should be securely riveted.

QUESTIONS ON CHAPTER IV.

i. Given a beam fixed at one end and loaded with a weight W tons at the other,
describe the system of forces acting at any section, neglecting the weight of the beam. If the
beam be 10 feet long and the load 2 tons, plot the diagrams of shearing forces and bending
moments, and give numerical values for a section 4 feet from the free end of the beam.

, _fS.F.,2tons.
* m - \B.M., 96 inch tons.

2. Referring to the previous question, if the given load be spread evenly over the beam,
indicate the forms which the curves of bending moment and shearing force will then take.

3. A beam 20 feet long supported at each end has a. load of 3 tons concentrated at a
point 2 feet from the middle of the length. Draw the diagrams of shearing forces and bending
moments, and indicate the value of the maximum bending moment.

Ans. — Max. B.M. = 172-8 inch tons.

4. Assuming the load in the previous question to be evenly distributed over the length
of the beam, calculate the maximum shearing force and bending moment, and indicate the
points in the length at which these are in operation.

^ ns f Max. S.F. = 1*5 tons acting at points of support

\Max. B.M. =90 inch tons acting at middle.

5. Show that diagrams of S.F. and B.M. may be derived by a graphic process, and
employ in your explanation the case of a beam fixed at one end and uniformly loaded.

6. Explain how to construct a curve of loads for a ship floating in still water, and state
what tests you would apply to prove the accuracy of your work.

7. What is the connection between curves of loads, shearing forces, and bending moments,
and show in one diagram the approximate forms these diagrams would take in the case of a
cargo vessel floating *' light" in still water.

8. A box-shaped vessel 200 feet long, 30 feet broad, 20 feet deep, floats in still water at
a draught of 10 feet. If the weight of the vessel be 1000 tons uniformly distributed, and if
there be a cargo of 715 tons uniformly distributed over half the vessel's length amidships, draw
the curves of S.F. and B.M. and state the maximum shearing force and bending moment

Ans _J Max - S - F - = *78'5 tons.
* ns ' \Max. B.M. = 8925 feet tons.

9. If the sides and top and bottom of vessel in previous question are composed of steel
plating \ inch thich, find the greatest stress to which the material is subject under a maximum
hogging moment of 8000 feet tons. Ans. — 1 '82 tons per square inch.

10. — Assuming a cargo steamer in loaded condition to be poised on the crest of a wave
sketch roughly the curves of loads, shearing force and bending moment.

11. Referring to the previous question, at what points approximately in the length will the
maximum shearing forces act and where will the maximum shearing stress intensity be developed?

12. Taking the box vessel of question 8, and assuming her to be under a maximum
shearing force of 400 tons, find the mean shearing stress over the section, and also the
maximum shearing stress. , /Mean shear stress = '66 tons per square inch.

\Max. shear stress = 1*82 ,, ,,

13. Enumerate the various local strains to which ships are liable, and the methods adopted
to strengthen the vessel against them.

CHAPTER V.
Types of Cargo Steamers.

NOT the least among the many important points to be settled by an owner
in deciding upon a new ship, is the question of type. A ship may be
suitable as to cost, may be strong enough, have good speed and dead-
weight capability, and yet may prove herself very unsatisfactory, if not an utter
failure, in certain trades. Every owner of experience is aware of this, and is
careful to see that he gets a ship suited to his purpose.

Nowadays, an owner who knows his requirements can usually get them
carried out in this matter. But this was by no means always the case. At
one time it seemed to be thought that ships must be built to certain fixed
designs, and cargoes had often to be adapted to suit a vessel's arrangements
rather than the latter being made to suit the former, resulting in much annoy-
ance, inconvenience, and expense.

With the expansion in oversea trade, however, but more especially with the
changes in materials ot construction — from wood to iron, and iron to steel —
and the progressive spirit of the age, came a gradual evolution of type, until
the cargo steamship of to-day has reached a high standard of excellence, and
where applied to special trades, has become almost the last word of efficiency
for the purpose intended.

The variations which have marked this evolution and brought cargo vessels
to their present stage of development have been, generally speaking, in the
following directions, viz. : — (i) in design of structure to provide hulls of degrees
of strength suitable for different trades ; (2) in form of immersed body and
in general outline and appearance ; (3) in disposition of materials ; (4) in
internal construction.

STRENGTH TYPES.— It was long ago recognised that for economical
working different cargoes should have different classes of vessels : that cargoes
of great density, for instance, which occupy little space in comparison with
weight, such as iron ore or machinery, and heavy general cargoes, should be
carried in strong vessels having great draught and displacement and limited
hold space ; and cargoes of less density, such as grain, cotton, wood, and light
general cargoes, should be accommodated in ships of relatively greater hold
capacity, but less deadweight capability. Thus, until their recent revision,
Lloyd's Rules provided special schemes of scantlings for three strength types,

76 SHIP CONSTRUCTION AND CALCULATIONS.

viz., three-deck, spar-deck, and awning-deck types. Of these, the first-named
was the strongest, and was reckoned to be able to carry any kind of cargo to
any part of the world on a greater draught than any other type of vessel of
equal dimensions. With regard to the spar and awning-deck types, we have
the authority of the late Mr. Martell, a former chief surveyor to Lloyd's
Register, for saying that they were not originally intended as exclusively cargo
carriers. The upper 'tween decks were really meant to accommodate passengers,
and the weather deck and the shell plating and side framing above the
second or main deck, were allowed to be of comparatively light construction.
But although thus built of smaller scantlings than the corresponding three-
deck vessel of same absolute dimensions, these lighter vessels were not of less
comparative strength. Their draughts were restricted, their loads reduced, and
hence also the leading movements acting upon them ; so that their thinner
materials were quite sufficient to ensure as low a stress per square inch on the
upper and lower works as in the corresponding three-deck type of vessel.

In the development of ship construction, the foregoing types have under-
gone modification. In Lloyd's latest Rules, only two distinct standard types are
mentioned, viz., the full scantling vessel, and an awning or shelter-deck type.
The latter has still the characteristics of other vessels of the class, namely, light
draught and large capacity, but has otherwise been greatly improved.

FORM TYPES.— With regard to changes of form, it must be admitted that
the body of the modern cargo steamer is no thing of beauty. The sentiment
which demanded fineness of form and grace of outline has passed away under the
pressure of ever-increasing competition. From the fine-lined under water bodies,
with displacement co-efficients of from *6 to 7, and the nicely rounded top-
sides of former days, we have come to sharp bilges, more or less vertical sides
and bluff ends, with displacement co-efficients ranging from '8 upwards. Cer-
tainly this side of the development of cargo ships has not proceeded on
aesthetic lines.

Appearances apart, however, and considering the case from a purely money-
making standpoint, the changes have been in the right direction. The re-
searches of the late Dr. Froude and others, and experience gained from actual
vessels, has shown that at moderate speeds like 8 or 10 knots — the speeds of
ordinary cargo vessels — the resistance to be overcome in propulsion is largely
due to surface friction, the element of wave-making resistance only becoming
important at higher speeds. As a considerable increase in displacement and
therefore in deadweight capability can be obtained by a moderate increase in
surface, the easiest and cheapest way for an owner to increase the earning
power- of his vessel is obviously to fill her out forward and aft, and this has
become the order of the day. Of course, for the best results, the filling out
process must be done with judgment. An expert designer can do much even
■with the fullest co-efficients. In general, vessels of '8 blocks and upwards
should have small rise of floor and relatively sharp bilges amidships, thus
. allowing most of the fining away to be done Lowards the extremities. In some

TYPES OF CARGO STEAMERS.

cases this method has not been followed. It should be said that in Lloyd's
former rules, the half girth appeared as a factor in calculating the numerals, and
this induced some builders, for the sake of getting lighter scantlings, to design
full cargo vessels with abnormally fine midship sections, thus causing the ends
to be very clubby. But such vessels when built invariably proved unsatis-
factory. They were found difficult to steer and therefore unmanageable in a
seaway, also harder to drive, than vessels of 'similar block co-efficients designed
on normal lines. Under the new rules the girth does not influence the
numerals, and there is now no temptation to design freak ships of the kind
mentioned; still, owners should not take too much for granted in ordering their
cargo "tramps," but should see that they get a maximum of good design with
any given conditions.

More striking than the changes of the under-water forms, and those which
have caused cargo vessels to be classified into various form types, have been those
due to the imposition of deck erections on the fundamental flush-deck steamer.

Very early in the history of the iron merchant ship, the necessity of
affording some protection to the vulnerable machinery openings led to the latter
being covered by small bridge erections. Then the obvious advantages of
having the crew on deck caused the accommodation for the latter to be raised
from below and fitted in a forecastle, this erection incidentally forming an
admirable shield from the inroads of head seas, and the release of the space
under deck making a desirable addition to the carrying capacity. Finally, poops
were fitted, experience showing the necessity of raising the steering platform
from the level of the upper deck. Thus the three-island type was arrived at,'
whose outlines are characteristic of many of the cargo steamers of to-day (see
fig. 68).

The next step in the development of deck erections was in the direction
of increased lengths, as, under Government Regulations, which became operative
in 1890, considerable reductions in freeboard could thereby be gained, and,
particularly as, provided they had openings in their end bulkheads, which,
however, might be closed in a temporary manner, the erections were allowed
to be exempt from tonnage measurement. Thus long bridges became common,
and eventually vessels were built with bridge and poop in one, making, with a
disconnected forecastle, one form of the well-deck type (see fig. 69).

Fig. 68.

Fig. 69.

e&e

SHIP CONSTRUCTION AND CALCULATIONS.

Fig. 70.

Fig. 71.

f |

£&6

Fig. 72.

i> 0j -°

The obvious advantage of having a continuous side and deck, and the
admirable shelter which the enclosed space would afford for cattle, etc., very
soon produced the suggestion to fill in the former gap between forecastle and
bridge; and this was rapidly carried into effect when it was found that by
having one or more openings in the deck with no more than temporary means
of closing, the space would escape measurement for tonnage. In this way the
shelter-deck type (see fig. 70) was evolved— a type in recent years much run
upon for large cargo vessels, and which, as previously mentioned, is now a
standard of Lloyd's rules.

Other modifications have consisted of short bridges on longer ones and on
shelter decks, but these can hardly be considered as constituting distinct types.

For the smaller classes of cargo carriers a somewhat special type of
steamer has been developed, familiar to all who take an interest in ships, as a
quarter-decker, which, in reality, is a one or two-decked vessel with the' main
deck aft raised (see fig. 71). This raising of the after deck was undoubtedly
due to considerations of trim. It was found that owing to the finer form aft
and the large amount of space taken up by the shaft tunnel, the tendency with
the normal deck line was to trim by the head when loaded, the predominance
of cargo at the fore end causing this. To correct this state of things the hold
space aft was increased by raising the deck.

While the quarter-deck has certain advantages, such as good trim and
general handiness, it has some drawbacks, one of which is the difficulty of
making up the strength sufficiently at the break of the main deck. The usual
plan is to double the shell plating and overlap the main and quarter deck
stringers in way of the break, the hold stringers at this part being also over-
lapped. In vessels of a size requiring a steel deck or part steel deck, the latter

TYPES OF CARGO STEAMERS. 79

is overlapped where broken to form the quarter deck, and the two portions
connected by substantial diaphragm plates. The doubling of shell and over-
lapping of stringers is also carried out.

The foregoing, or something equivalent, is what must be done to make
good the loss of continuity. It is seen to involve a considerable amount of
bracketing and troublesome fitting work, which tends to raise the first cost of
the vessel. In the vicinity of the break, too, there is much broken stowage
space; yet, in spite of all, for some trades this type still remains a strong
favourite.

Another modified type, in some respects the opposite of the last in that
it leads to an increased hold capacity forward over the normal type, is the
partial awning-decker. It is to be supposed that with ordinary cargoes this
type would trim badly, but it appears to have been found very suitable for
special light bulky cargoes. It was at one time very popular, but of recent
years has not been much in evidence. The external appearance of the partial
awning-decker is shown in fig. 72. It is seen to be a quarter-decker with the
forward well filled in, and the precautions already described for maintaining
the strength at the break have also to be taken in this case.

One clear consequence of the long erections now become prevalent is un-
doubtedly the modern system of distributing the materials of construction.

Bridges which are very short have small structural value, as they are not
really part of the hull proper, and should not be considered in estimating the
longitudinal strength. It is otherwise, however, with bridge erections of
substantial lengths, which must withstand the structural bending stresses acting
on the vessels of which they form part. Moreover, it follows from the principles
expounded in the previous chapter, that the heaviest longitudinal materials
should be placed at the deck, stringer, and sheerstrake of an erection, whether
it be a long bridge, an awning or a shelter deck, as the moment of inertia of
the material at a section is thereby increased, and the stress under a given load,
which is of maximum value at these parts, reduced. Modern vessels are now
required to be built in this way by the rules of Lloyd's Register and of the
other classification bodies, the old practice of making superstructures of light
build and massing the strength at the second deck from the top being dis-
continued. This may be considered to mark an important advance in the
scientific construction of ships.

CONSTRUCTION TYPES.— Coming now to the changes that have taken
place in the internal construction of vessels, we find these to be of a wide-
reaching character. Fig. 73* is the midship section of a large passenger and
cargo steamer as built 25 to 30 years ago, and illustrates all the characteristics
of the time, viz., thin side framing, numerous tiers of beams, ordinary floors,
and deep hold keelsons. The expansion of commerce, however, the opening

* See an interesting paper on "Structural Development in British Merchant Ships," by
Mr. J. Foster King, in the Transactions of the Institution of Naval Architects for 1907, to
which the author is indebted for particulars in preparing some of the sketches in this chapter.

SHIP CONSTRUCTION AND CALCULATIONS.

up of new trades, and the specialising of vessels for these trades, led first to
increase in the average size of vessels, then to various modifications in their
internal economies. The trouble and expense attending the use of dry ballast
led to the adoption of water ballast tanks, which, ultimately becoming
incorporated in the structure, caused the disappearance from the holds of the
huge plate side girders which, as shown in fig. 73, accompanied the fitting of
ordinary floors. In a later chapter we shall deal in detail with ballast tanks,
but their general design and arrangement may be gathered from figs. 74 to 85.
An early modification in the structure was the deepening of the holds by the

Fig. 73.

suppression of the lowest tier of beams, required by the construction rules of
the time, and the fitting at every fifth or sixth frame of plate webs having face
bars on their inner edges, the hold stringers being deepened to come in line
with the inner edge of the plate webs, and the whole forming a strong box-like
arrangement which amply made up the deficiency caused by the omission of
the hold beams (see fig. 74). This style of construction long remained in favour
and is still sometimes preferred, but the loss of stowage capacity, particularly
for case cargoes, eventually led to its general abandonment in favour of the
deepening of the frame girder itself, the system of framing which in one form
or another is found in the cargo steamers building in the yards to-day.

TYPKS OF CARGO STEAMERS,

SHIP CONSTRUCTION AND CALCULATIONS.

A natural development which came, although not quite immediately, was
the reduction in size of the hold stringers, which, as stowage breakers, were
found not less obnoxious than the webs. Moreover, the deep side framing
alone was sufficient for all the demands of local stresses, and owing to their
proximity to the neutral axis, the extent to which these hold stringers assisted
me ship against bending was comparatively trifling. In fig. 75 is shown the
hold stringers of fifteen years ago, and in fig. 76 those of the present day.
Their work now is to keep the frames in position, to prevent them side
tripping, and to stiffen the shell between the frames.

Fig. 76.

330 Feet Steamer.

Recent experiments made by Lloyd's Register have gone to show that
up to a frame depth of 7 inches (the limit of the experiments) there is no
tendency to side tripping, and since then vessels have been built with a re-
duced number of hold stringers, and in a few recent cases with none at all.
Whether the hold stringer will ultimately disappear from the modern ship
as an element of construction remains to be seen. This, it may be said,
is the view taken by some naval architects, but the general feeling seems
to be in favour of its retention in a modified form.

Improvements in the manufacture of steel sections in recent years, and
the broad-minded view now taken by the classification societies, have made

TYPES O* CARGO VESSELS.

it possible for builders, following the line of simplification of parts, to still
further satisfy the demands of shipowners for large holds clear of beams,
stringers, and numerous hold stanchions. Hence has come the well-known single-
deck type {see fig. 77) Vessels of a size ordinarily requiring, by Lloyd's former
rules, three tiers of beams and two steel decks, have been built with a
single steel deck and one tier of beams, the structural strength, transverse
and longitudinal, being made good by deepening the side frames and in-
creasing the scantlings of the deck, shell plating, and double bottom- Purely

Fig. 77.

350' 0" X 51' 0" x 2ff 0".

, LINT 0f_BBI0CEJ)KK

single-deck vessels have gone on increasing in size until they have attained
lengths of 350 feet, and depths exceeding 28 feet, and it appears likely the
advancement will still proceed so long as the needs of commerce demand
it. In Lloyd's latest rules, the construction of single deckers up to a moulded
depth of about 31 feet is provided for, but so far as we are aware, no
single-deck vessel of ordinary design has been built approaching this depth.

Fig. 78 illustrate? a type which may be considered to be in the tran-
sition stage towards the pure single decker. It has bulb angle framing

.84

SHIP CONSTRUCTION AND CALCULATIONS.

and strong hold beams widely spaced in association with arched webs and
a broad hold stringer. Many vessels of this type have been built on the
N.E. coast and have proved highly satisfactory. The designers and first
builders of this type are an important Wearside firm.

With the removal of hold stringers and beams, the presence of numerous
hold pillars became specially objectionable. A middle line row for most
trades is perhaps no great drawback, but with the increased breadths at-

Fig. 78.

SS. 350' 0" X 49' 0" X 28' 0".

Lig£0F_8RUCE_DECK

tendant on the steady rise in general dimensions, now the order of the
day, additional rows of stanchions between the middle and the side, with
the ordinary construction, became imperative. For a time, and up to a
certain point, the case of vessels with breadths beyond that at which quarter
stanchions are necessary, was met, without resorting to the latter, by increas-
ing the scantlings of the beams and ot the middle row of pillars, but a
limit was soon reached, and the question of the omission of pillars had to
be reviewed from other standpoints. Hence arose the system of fitting wide

TYPES OF CARGO STEAMERS.

spaced strong pillars in association with deck girders. Centre line rows of
closely spaced pillars with one or two quarter pillars in each hold are now
fonnd in vessels of 50 feet breadth and upwards. In some cases the
centre row has been omitted, the whole work being done by, say, four
specially heavy pillar columns in each hold.

The great convenience of the latter arrangement from a stowage stand-
point can readily be conceived, and although it entails a considerable addition
in cost over the common arrangement, many shipowners have adopted it.

Fig. 79.

SS. 340' 0" x 45' 6" x 27' 3" and 34' 3".

A few vessels have been built so as to be able to dispense with pillars
of any kind, and to these we shall refer presently

SPECIAL TYPES.— Besides the types of vessels already described which
may be considered the standard ones, there are others of quite distinct char-
acter, which the needs of commerce, the enterprise of shipowners, and the
genius of shipbuilders, have called into being. Of these, probably the most
important is the well-known turret-deck type of Messrs. Doxford. Fig. 79
shows the midship section of one of these vessels, and illustrates the striking
differences between them and those of ordinary form.

SHIP CONSTRUCTION AND CALCULATIONS.

The principal departure is in the outward form at the topsides, which,
instead of being carried up with a moderate tumble home, are curved inwards,
forming a central trunk or turret. The working platform is on the top of this
turret, which runs forward and aft and contains all hatches, deck machinery,
derricks, and everything requisite for efficiently working the vessel.

The internal framing of the majority of these vessels (see fig. 79) is on
the wide-spaced hold beam and web-frame system ; but in recent cases the
no hold obstruction principle has been carried out, hold beams and pillars
being entirely omitted, and the strength made good by fitting deep web-plates

Fig. 80.

SS. 350' 0" x 50' 0" x 26' 3" and 33' 6".

with attachments to the turret deck, ship sides, and tank top, as shown in

fig. So.

Among the advantages claimed for this type over the ordinary ones are
its self-trimming qualities, which make it well suited for bulk cargoes ; the
greater safety which it affords to all vulnerable openings, such as hatches
ventilators, etc., owing to the turret being much higher than the ordinary
weather deck ; its greater stiffness and longitudinal strength, owing to its shape *
increased depth and better distribution of longitudinal materials, the latter
circumstance making it possible to reduce the structural weight and thus in-

TYPES OF CARGO STEAMERS.

crease the deadweight. Although it cannot be said that these vessels have a
nice appearance, it must be admitted that they have been a long time in
service, and seem to be increasing in popularity as purely cargo boats.

Another type, of which there is now a considerable number afloat, is
Messrs. Ropner's patent trunk steamer. This class is of normal single deck
construction to the main or harbour deck; above this there is a central trunk
running fore-and-aft; the top of the latter forms the working deck and is
fitted with hatchways, winches, etc. The ship is kept in form by strong beams

Fig. 81

SS. 350' 0" x 60' 0" x 25' 3" and 33' 3".

at the hatchway ends, and the trunk is stiffened by webs and supported by
strongly built centre stanchions. This ship, like the turret design, is specially
suitable for bulk cargoes like grain, the trunk forming an admirable self-trimmer
(see fig. 81).

The Dixon & Harroway patent ship is another type whose speciality is its
self-trimming arrangements. In this vessel (see fig. 82) the upper corner of the
hold is plated in, the main frame of the vessel being carried up in the hold
space. This corner space is well adapted for ballasting purposes, the high
position of the ballast conducing to steadiness in a seaway. This type is of

SkiP CONSTRUCTION AND CALCULATIONS.

Co Em
CO N

Co «

types of cargo steamers.

3 9

great longitudinal strength and is also well suited to resist the tendency to
transverse change of form set up when a vessel is labouring in a seaway. Self-
trimming also forms the chief claim to distinction of the vessel shown in fig.
83. It is seen to resemble the last type somewhat with the corner tanks away;
and on the latter account is not so efficient from a strength standpoint.
As in the Ropner trunk vessel, the ship is worked from a central fore-and-
aft platform.

Still another variation of the trunk or turret type is that devised by Mr.
Henry Burrell. Like the other vessels just referred to this one is a self-

Fig. 84.

SS. 305' 0" x 46' 9" x 24' 0" and 30' 3".

trimmer, and, as well as the upper trunk, has the corners at the bilges filled in
(see fig. 84) and the inner surface sloped towards the centre, thus obviating the
broken stowage space which might otherwise occur at the bilges. Incidentally,
the corners thus cut off from the holds form a desirable addition to the
ballast capacity. The deck, sides and trunkways are supported by cantilever
webs, and there are no hold pillars.

Other special types have been built, or are building, differing more or less
from the foregoing, but in general not sufficiently to make it necessary to refer
to them. One design, however, that of Mr. Isherwood, is of such distinctive
and interesting a character as to warrant its being singled out. This type is

SHIP CONSTRUCTION AND CALCULATIONS.

framed on the longitudinal system, and in this respect recalls that famous
work of Scott Russell and Brunei— the Great Eastern. Like the earlier vessel,
the new ship has main frames and beams running fore-and-aft, with widely
spaced transverse partial bulkheads. There is, however, no double skin on the
sides, the inner bottom being of the normal present-day type, except that the
main internal framing is longitudinal instead of transverse.

Fig. 85 shows the midship section of a medium-sized cargo steamer
framed on this system. The longitudinal beams and frames are seen to
consist of bulb angles at wider spacing than on the transverse system. It
should be noted, however, that the settlings of the frames are gradually

Fig. 85.

increased towards the bottom of the vessel, where they have to withstand
greater loads, the intensity of the water pressure increasing in proportion to
the depth below the surface.

The transverse strength is made up by strong transverses or partial
bulkheads attached to the shell-plating between the frames, and stiffened on
their inner edges by stout angles. The transverses are spaced from about
12 feet to 16 feet apart in ordinary cases, according to size of vessel, the
largest vessels having the closest spacing.

The double bottom, as previously mentioned, has fore-and-aft continuous
girders, with intercostal transverse floors in line with the tranverses and also
midway between them, the latter being required to provide sufficient strength

TYPES OF CARGO STEAMERS.

9 1

for docking purposes and to resist the excessive stresses which come on the
bottom through grounding.

It is claimed for this type of vessel, several samples of which are now
afloat and giving good accounts of themselves, that it has greater strength
and less relative weight than the normal type. The saving in weight is a
point of great importance, as apart from any reduction of first cost which
this may represent, it means for the vessel greater deadweight capability
and therefore increased earning power.

Fig. 86.

The Isherwood system of construction appears to be specially suitable for
oil vessels.* Fig. 86 is a section of an oil steamer framed in this way, the
dimensions of which are, viz.: — length, 355 feet; breadth, extreme, 49 feet, 5
inches; depth at centre, 29 feet. The longitudinal frames from the deck to
the upper turn of the bilge are bulb angles as shown; on the bottom they
are built of plates and bars ; the spacing is 29 inches. The beams are bulb
angles spaced 27 inches apart. The main oil tanks are 30 feet long, and
two strong transverses are fitted in each tank between the boundary bulk-

*See a paper by Mr. Isherwood in the T.I.N. A. for 1908, from which figs. 85 arul 86 are taken.

9 2 SHIP CONSTRUCTION AND CALCULATIONS.

heads. The transverses are fitted to the shell-plating between double angles
and have heavy double angles on their inner edges.

The longitudinal frames and beams and longitudinal stiffeners on middle
line bulkhead are cut at the transverse bulkheads and efficiently bracketed
thereto in order to maintain the continuity of strength. In way of the
double bottom, which is fitted for a portion of the vessel's length amidships,
alternate transverses are fitted continuously around the bottom to the middle
line, and the longitudinal girders are fitted in long lengths between these
transverses, and efficiently attached thereto. The remaining transverses are
stopped at the deep girder in the double bottom next the margin-plate, and
are then fitted intercostally between the longitudinals to the centre line. The
margin-plate is fitted intercostally between the transverses, and connected to
them by double-riveted watertight collars.

A comparison of the longitudinal stress acting on the bridge gunwale
amidships of this vessel with that acting on an oil vessel of the same
dimensions built on the ordinary system, showed the former to be 18J per
cent, less than the latter. In spite of this there is stated to be an
estimated saving in weight of materials, under the new system, of 275 tons.

CHAPTER V.

Practical Details.

KEELS AND CENTRE KEELSONS.— The keel may be considered the
foundation of a ship's structure. The simplest form of keel fitted in iron
or steel vessels consists of a forged bar running almost the full length of
the vessel. At the ends it is scarphed into the stem and sternpost, the
three items together forming a complete longitudinal rib. The bars forming

Fig. 87.

RIDER PUTE

STRAKX

the keel are fitted in lengths averaging about 40 feet, joined together by
vertical scarphs nine times the thickness of the keel in length. These scarphs
are frequently riveted up previous to the fitting of the shell by means of small
tack rivets, so as to allow the keel to be faired. There is an objection to the
use of tack rivets, in that, if it be necessary to remove a keel length,

SHIP CONSTRUCTION AND CALCULATIONS.

say, for repairs after grounding, plates on both sides of the keel have
to be removed in order to knock out the tack rivets ; for this reason
they are sometimes omitted. The main rivets connecting the scarphs together,
and also the keel to the shell, are of large diameter, spaced five diameters
apart, centre to centre, and arranged in two rows, usually chain style, as in

Fig. 88.

fig. 87, although zig-zag riveting is occasionally adopted; in the latter case,
care must be taken to keep the rivets clear of the garboard strake butts.

It will be seen by referring to fig. 8 7 that the only connection that
this keel has to the main structure is through the riveted connection to the
garboard strakes. For this reason it is frequently called a hanging keel.

Fig. 89.
BAR HvEEL : INTERCOSTAL CENTRE KEELSON

RIDER PLATE

The simple bar keel is sometimes fitted in association with a centre keelson
running along the tops of the floors, consisting of double bulb angles in small
vessels, and a vertical plate and four angles, two top and bottom, in larger
vessels. In the largest vessels, a rider plate is fitted on top of the upper angles
and a foundation plate on top of floors below lower angles. This style of keel-

KEELS AND CENTRE KEELSONS.

son, which is depicted in fig. 87, but without a foundation plate, is seen to
have no direct connection with the external keel. From what we already
know of the bending of beams, we must see that the arrangement is by no
means a perfect one. Bending separately, the keel and keelson do not offer
the same resistance as if rigidly joined. Moreover, the floors lying at right
angles to the line of stress give no support, but develop a tendency to trip,
as shown exaggerated in fig. 88. The weaknesses pointed out in the above
plan may be largely corrected by fitting plates between the floors, from the

Fig. 90.

FOUNDATION
PLATE -,_

1 |o

jo 1 I

I — — —Ok

ha 1 \

THRO PLATE

keel to the keelson (fig. 89). This transforms the separate beams of small
resisting power into one powerful girder. The intercostal plates, too, prevent
any possibility of movement of the floors.

A better arrangement than the preceding one consists of a continuous
centre through plate, extending from the top of keel to the top of floors, or
top of keelson (see fig. 90). Sometimes the centre plate is extended to the
bottom of the keel, the required thickness of the latter being made up by
means of two side bars (see fig. 90a). These arrangements, of course, entail

RIOER PLATE

FOUNDATION
PLATE ^

KEEL SIDE BARS -

the severing of the floors at the middle line, causing considerable reduction
in the transverse strength ; the floors are, however, connected to the centre
girder by means of double angles, and a flat plate 12 inches wide, and of
.the same thickness as the centre through plate, is fitted on top of the
floors each" side of the centre plate, and connected to the latter by angles or
bulb angles, thus making good the transverse strength and adding to the longi-
tudinal strength. In the largest vessels the keelson is run up high enough
to take four angles with a rider plate on the upper two (see fig. 90a).

9 6

SHIP CONSTRUCTION AND CALCULATIONS.

It will be observed that a practical difficulty crops up in the riveting
of the keel to the garboard strakes, in the case of a side bar keel, as five
thicknesses of plating require to be united by the same rivets. There are two
rows of such rivets, of size and spacing similar to the bar keel, and as it is
usual to punch these holes before fitting the plates, it can be imagined that
very careful workmanship is needed to keep the rivet holes concentric. As a
matter of fact, they are frequently more or less obstructed. In such cases,
before proceeding with the riveting, the holes should be rimered out. The
objectionable plan of drifting partially blind holes — that is to say, of driving
a tapered bar of round steel or drift punch into them, so as to clear a
passage for the rivet — should not be encouraged. It is known, and has been
proved many times in practice, that the bruising which the material round
the edges of the holes gets by drifting, renders it brittle and therefore liable
to break away, loose rivets resulting in consequence.

An objection common to all projecting keels is the increase of draught
which they entail. It is always considered a good feature in a vessel, and

Fig. 91.
FLAT PLATE KEEL

INTERCOSTAL CENTRE KEELSOM

FLOOR

INTERCOSTAL'

particularly in a cargo vessel, to have a moderate draught of water. The reason,
of course, is that many ports will be open to a vessel, if of shallow draught,
which would otherwise be closed. These considerations have led many owners
to adopt what is called a flat-plate keel in preference to the one we have been
dealing with. In this case, the ordinary shell-plating is continued under the
vessel instead of being stopped on each side of a projecting keel ; the middle
line strake is increased somewhat in thickness, and is considered to be the
keel of the vessel (see figs. 91 and 91a). This horizontal plate would of itself
be a very inefficient substitute for the rigid vertical bar of the ordinary
keel, but it is usually fitted in conjunction with an intercostal or continuous
vertical centre plate, the two being connected together by double angle bars.
With an intercostal centre plate, the floor plates are continuous ; with a con-
tinuous centre plate, they are severed at middle line, and abut against the
centre plate on each side. In both cases the floor and centre plates are
connected by double vertical bars ; this prevents any movement of the parts.
It should be noticed that with a flat-plate keel, the rolling reducing property
of the projecting keel is lost. It is the custom, however, in modern caro-o

KEELS AND CENTRE KEELSONS.

vessels, to make up for this by fitting longitudinal bars or rolling chocks
at the bilges (see figs. 75, 76, etc.).

The flat-plate type of keel is frequently fitted where there is a double
bottom (fig. 91b). As the floor plates are then of considerable depth, a much
more satisfactory connection with the centre plate is obtainable than with
ordinary shallow floors; by Lloyd's Rules double angles are not required in this
case, except in the machinery space, where they are always necessary, until
the transverse number reaches 66, corresponding to a vessel say, 300' x 40' x 26,
when double angles are required for half length amidships.

Fig. 91a.

CENTRE THRO PLATE KEELSON

CENTRE THRO PLATE

Fig, 91b.
CONTINUOUS CENTRE GIRDER - DOUBLE BOTTOM

1^1

—

fc • O O O | C "^0 O °l

f o o~~J

oa —

OOOI o

o|
000'

e o,

000'

, °

ol °

\ °;

' "

\' \° ° °\ ° 1

1° ° °

LLOYD'S NUMERALS. — In the previous paragraph reference has been
made to Lloyd's Rules, and we now propose briefly to consider the methods
adopted in these Regulations for the construction of ships, of assessing the
scantlings of the various parts of a vessel. Lloyd's Rules in this particular differ
in details from the Rules provided by other classification bodies, but for the
purpose of illustration it may be sufficient to refer to them alone, particularly
as they represent the common practice of present-day shipbuilding. The
numbers under which the Tables of Scantlings are graduated, are derived
from the dimensions ; it is therefore necessary to define these. The definitions
given in Lloyd's Rules are as follows ; —

SHJP CONSTRUCTION AND CALCULATIONS.

Length. — The length (L) is to be measured from the fore part of the
stem to the after part of the sternpost on the range of the upper-deck
beams, except in awning or shelter-deck vessels, where it is to be measured
on the range of the deck beams next below the awning or shelter deck.

Breadth. — The breadth (B) is to be the greatest moulded breadth of
the vessel.

Depth. — The depth (D) is to be measured at mid-length from the top
of keel to top of beam at side of uppermost continuous deck, except in
awning or shelter-deck vessels, where it may be taken to the deck next
below the awning or shelter deck, provided the height of the 'tween decks

Fig. 92.

AWN 1 INC OR SHELTER PSjCK OR BRlOCEQECK

does not exceed 8 feet ; B and D are indicated in fig. 92, which shows an
outline midship section of a vessel.

From these dimensions the scantling numbers are obtained thus : —
Transverse number = B 4- D
Longitudinal number — L x (B 4- D).
The transverse number regulates the frame spacing and the scantlings of
the floors. Thus, taking a vessel of 45 feet breadth and 28 feet depth,
we have —

Transverse number = 45 4- 28 = 73.

And under this number we find in the appropriate Table of the Rules that
the frame spacing should be 24 J- inches, and the floors 30 inches deep at
middle, '46 of an inch thick for ■? length amidships, tapering to -38 of an
inch at ends.

Lloyd's numerals.

The scantlings of the frames are governed by the transverse number,
i.e., by the size of the vessel, and also by the extent to which the frame
is unsupported. The frame is assumed to be supported at the first tier of
beams above the base and at the bilge. Reverting to fig. 92, d is the
unsupported length of frame. Two cases are indicated, one assuming a tier
of beams to exist below the upper deck, another assuming the frame to be
unsupported from the bilge to the upper deck. It will be observed that
at the bilge d is measured from a line squared out from the tank at side.

Fig. 93.

In this case there is an inner bottom ; when a vessel has ordinary floors,
the line is squared out from the height of the floors at middle.

The rules provide scantlings of frames for values of d up to 27 feet,
this figure apparently marking the limit of a purely single-deck vessel. In
fig. 93 the framing of three single-deck vessels of different dimensions is
given, and shows how the scantlings increase with increase in size of vessel.
The longitudinal number regulates the scantlings of the keel, stem, sternpost,
side and bottom plating, double bottom, side stringers, keelsons, lower deck

IOO SHIP CONSTRUCTION AND CALCULATIONS.

stringer plates, and lower deck plating. It is also employed with a number
giving the proportions of length to depth in fixing the scantlings of the
upper works.

The importance of distribution of materials and depth of girder, pointed
out in Chapter IV. } is fully recognised in the Rules. Thus the heaviest
materials are concentrated at the side and deck-plating of upper, awning, and
shelter decks, and of long bridges. Also the scantlings at these parts are
less in a deep vessel than in one that is proportionately shallow.

The depth employed in obtaining the proportions of length to depth for
use with the Tables is to be measured at the middle of the length from the
top of keel to the top deck at side in all cases, except in way of a short
bridge, when the depth is to be taken to the upper deck, which thus becomes
the strength deck. The scantlings at the upper deck beyond the ends of a
long bridge, are to be determined by taking the depth for proportions to the
upper deck.

Shallow vessels, which have lengths equal to or exceeding 1 3 J depths,
taken to the upper deck, are required to have a bridge extending over the
midship half length, or compensation in lieu. As the bridge deck becomes
the strength deck, this means a substantial increase of the depth of the
ship girder. In the case of still shallower vessels, namely, those having
lengths exceeding 14 depths, the question of the longitudinal strength has
to be carefully considered, and Lloyd's Committee require proposals to be
laid before them.

FRAMES. — Next to the keel the transverse frame -is probably the most
fundamental part of. a ship's structure, especially in vessels with ordinary floors.
As previously explained, it extends from the keel to the top of the vessel in
a transverse plane, and gives the form of the ship at the point at which it
is fitted (see figs. 37 and 74). Frames of vessels built to Lloyd's Rules may
be spaced from 20 to 2>Z inches apart, according to the size of vessel. In
special cases, the spacing may exceed 33 inches, if suitable compensation be
made. At the fore end, from a fifth the vessel's length from the stem to
the collision bulkhead, owing to the pounding stresses to which this part of
the vessel is liable at sea, the frame spacing should not exceed 27 inches,
unless the frames are doubled to the lowest tier of the beams. In the peaks
the frame spacing should not be greater than 24 inches.

Each complete transverse frame may be made up of two angle bars, i.e. % a
frame and reversed frame, as described in Chapter III.; or it may consist, as
in many modern cargo steamers, of a single angle or bulb angle ; or it may
be of channel section, with the addition, in the case of a large vessel, of a
reversed angle. Lloyd's Rules provide tables of scantlings of frames of these
various styles. In fig. 94 the side framing required for the vessel marked A
in fig. 93 is shown, the three equivalent types being indicated.

The fore-and-aft flange of a frame is riveted to the shell-plating, and
the transverse flange, in vessels having ordinary floors, is at its lower
part attached to a floorplate. When the construction consists of a frame

FRAMES.

101

and reversed frame, the latter is riveted to the frame on the sides of the
vessel, to the turn of the bilge, whence it sweeps along the top edge of
the floor, which being thus stiffened at top and bottom, becomes an efficient
transverse girder. Both the frames and reversed frames are usually butted at
the centre line, covering angle bar straps being fitted. The frame butt-straps,
or heel-pieces, as they are called, are usually about 3 feet long, and are placed
back to back with the frame, the floor-plate being between. These heel-pieces
should be so fitted as to bear on the top of the keel, when of simple bar type,
as in this way stresses due to docking or grounding are communicated

Fig. 94.

directly to the framing; .without unduly straining the rivets connecting the keel
to the garboard strakes. Heel-pieces are only fitted for three-quarters the
vessel's length amidships, the form at the ends making them unnecessary.
Where the middle line keelson is a centre through plate, the heel-pieces are
not usually fitted; and where the former is associated with a flat-plate keel it
is, of course, impracticable to fit them.

At the decks, the framing on each side of the vessel is connected by
cross beams, special attention being given to the beam-knee connections, as
the combination of beam, frame, shell-plating and deck-stringer in this neigbour-

102

SHIP CONSTRUCTION AND CALCULATIONS.

hood is most efficient for resisting the transverse racking stresses to which, as
we have seen, a vessel may be subjected when rolling among waves at sea.

WEB FRAMES. — When a transverse rib consists of a deep plate with
stiffening angles on its inner edge, it is known as a web frame. Lloyd's
Rules permit a system of web frames at six frame spaces apart, with com-
paratively light intermediate frames, to be substituted for the heavier frames
of the ordinary frame table, provided a deck be laid on the tier of beams at
the height d. In fig. 95, the largest of the three vessels indicated in fig. 93
is shown with web frames. It will be seen that the angles connecting the

Fig. 95.

SECTION SHOWINt
WEB FRAME

SECTION SHOWING
INTERMEDIATE FRAME

'iNTERWlDiATCl
TfUMES /
6V5V-4ZB-A

webs to the shell-plating and to the side stringers are of the same thickness
as the webs, and that angles are fitted to the inner edges of the webs and
stringers. When webs become of considerable depth, they are really partial
bulkheads, and to develop their full efficiency should have a substantial
shell connection. Thus webs 24 inches and above, in vessels built to
Lloyd's Rules, require double angles to the shell-plating, or equivalent single
angles double riveted.

A web frame being held rigidly at the deck by its connection to the
beams, and at the bilge by its floor or tank side connection, forms a

WEB FRAMES. 103

girder of comparatively short span, at least when compared with the side
stringers, which are only properly held at the bulkheads. For this reason
it is advisable to make the web frames continuous and the side stringers
intercostal, and this is usually done. At the junction of each web and
stringer, the discontinuity of the latter is made good by a double angle
connection to the webs, and by fitting a stout buttstrap to the stringer face
bar (see fig. 95). Web frames are attached by bracket knees to beams at
their heads, the knees being double riveted in each arm and flanged on their
inner edge. At the lower part, when associated with an ordinary floor, the
inner edge of the web frame is swept into the top edge of the former, the
connection to the floor being an overlapped riveted one. When the con-
nection is to be made to an inner bottom, it should consist of a riveted
angle on to the margin plate, with, in addition, a substantial gusset plate or
angle bar from the top bar of the web on to the inner bottom plating
(see fig. 95.)

FLOORS. — These vertical plates will be observed to have a maximum
depth, governed by the size of vessel, at the middle line where the transverse
bending stresses are greatest. Thence they gradually taper towards the sides,
the depth at three-quarters the half breadth, measured out from the middle
line on the run of the frame, being half that at centre line. From this point,
the upper edge of each floor sweeps into the line of the inside of the frame,
terminating at a height from the base line equal to twice its depth at the
middle line. There is one such floor at each frame, the floor, frame, and
reversed frame forming, indeed, a transverse girder, which is the most char-
acteristic feature of a vessel built without an inner bottom. Except in small
vessels, the floor-plates are fitted in two pieces, connected by an overlap or
buttstrap at the middle line, or alternately on each side of that line. When
a vertical through centre-plate is fitted, the floors are fitted close against it
on each side, a riveted connection being made by double vertical angle bars,
as shown in fig. 90. The loss of transverse strength due to cutting the
floors is also partly made good by a horizontal keelson plate fitted at the centre
line on top of floors, referred to when dealing with centre keelsons. When
inner bottoms are fitted, this part of the structure undergoes considerable
modifications, as we shall see presently.

SIDE KEELSONS. — As well as the centre keelson, vessels with ordinary
floors have keelsons midway between the bilge and the middle line. The main
function of these keelsons being to keep the frames and floors in their cor-
rect relative positions, intercostal plates are fitted between the floors and
connected to the shell-plating and to double angles on top of the floors.
These intercostal plates need not be connected to the floors, but in order
to develop their full efficiency should be fitted close between them. In
small vessels, under 27 feet breadth, one side keelson is considered sufficient;
in larger vessels, 27 feet and under 50 feet in breadth, two are necessary.
In fig. 96, side keelsons for vessels of various sizes are shown.

to4

SHIP CONSTRUCTION AND CALCULATIONS.

BILGE KEELSON. — In vessels of 50 feet and under 54 feet breadth,
in addition to two side keelsons, a bilge keelson is required on each side,
and should be carried as far forward and aft as practicable. This keelson,
like a side keelson, should have an intercostal plate connected to the shell-
plating (see fig. 96).

Fig. 96.

SIDE STRINGERS.— Between the bilge and the deck beams the framing
is tied together, and stresses to some extent distributed by stringers con-
sisting in small vessels of single angles riveted to reversed frames and lugs,
and in larger vessels of similar angles associated with an intercostal plate
connected to the shell-plating. Lloyd's Rules require side stringers of this
latter type in vessels of all sizes. According to these Regulations, the number

Fig. 97.

SECTION AT A.B.

~^r~

7 \_y w ^u s^7~

of side stringers depends on the value of d. When this is 7 feet and less
than 14 feet, that is in very small vessels, one is sufficient; where d is 14
feet to 21 feet, two are necessary; and when d is 20 feet and under 27
feet, three should be fitted.

All keelson and stringer plates and angle bars, when continuous, should

BEAMS.

105

be fitted in long lengths, and to obviate any sudden discontinuity of the
strength, adjoining butts should be carefully shifted from each other. Both
plates and bars should be strapped at the butts, the angle-bar straps con-
sisting of bosom pieces of the same thickness as the keelson bar and two
feet long, having not less than three rivets on each side of a butt (see fig. 97).

BEAMS. — A tier of beams is always fitted so as to tie the top of the
frames together and support the deck ; the functions of beams as elements of
strength in the structure of a ship are generally as have been already de-
scribed (see pages 71, 72).

The number of tiers of beams required in any vessel is a question of
transverse strength, but it also depends on the trade for which she is intended.

Fig. 98.

ELEVATION SHOWING STRINGER FACl ANCLE.,

Passenger boats usually require one or more decks below the upper one for
purposes of accommodation. Many cargo vessels, owing to the nature of their
cargo, also need one or more 'tween deck spaces; in most cargo boats, how-
ever, as has been said elsewhere, the desire is for deep holds, clear of beams
or other obstructions. Lloyd's latest Rules allow considerable scope to the
designer in this matter. As was seen in a previous paragraph, they allow
him, up to a certain point, to design his vessel entirely clear of beams below
the upper deck, if he so wishes. The value of tf, in such a case, will, of
course, be relatively great, and the scantlings of the side framing relatively
heavy (see fig. 93). This is the penalty exacted for unobstructed holds, and
it is obviously a just one; for each frame is a girder, whose strength is
governed by its span.

Io6 SHIP CONSTRUCTION AND" CALCULATIONS.

Spacing of Beams. — According to Lloyd's Rules, a complete tier of
beams, such as is required to form the upper point of support of the frames
in regulating the scantlings of the latter, may consist of : —
(i) Beams at every frame.

(2) Beams at alternate frames.

(3) Beams widely spaced up to 24 feet apart.

In arrangements (2) and (3) the beams are heavier than in (1) ; and
in (3) a broad stringer must be fitted on the ends of the beams with a
heavy face bar, and large horizontal gussets must be fitted between the
stringer and the beams (see fig. 98). Lloyd's Rules require beams to be
fitted at every frame in the following places : —

(1) At all watertight flats.

(2) At upper decks of single-deck vessels above 15 feet in depth.

(3) At unsheathed upper decks, when a complete steel deck is required

by the Rules.

(4) At unsheathed bridge, shelter and awning decks.

(5) At upper, shelter, or awning decks in vessels over 450 feet in

length, whether the decks be sheathed or not. Under erections,
such as poops, bridges, and forecastles in vessels less than 66 feet
in breadth, upper deck beams may be on alternate frames, except
for one-tenth the vessel's length within each end of a bridge, where
they are to be fitted at every frame.

(6) At the sides of hatchways, including those of engine and boiler

openings, in all unsheathed steel or iron decks.

Elsewhere, deck beams may be spaced two frame spaces apart, but
only if the frame spacing does not exceed 27 inches.

It is easy to grasp the reason for close spacing the beams on unsheathed
decks. With thin decks and widely spaced beams, the plating would probably
sag between the latter, which would make the decks most unsightly; with
beams at every frame the sagging tendency will be very slight. The close
beam spacing in way of hatches is necessary in view of the heavy weights
which may be brought upon the deck there during loading operations.

When a wood deck is laid, beams may be at alternate frames (except as
above stated in vessels over 450 feet). In this case, the steel deck is sup-
ported between the beams by the wood deck, the deck fastenings for the
wood deck being fitted between the beams. The beams forming the weather
decks are usually cambered, so as to throw off water quickly ; those of the
lower tiers are sometimes cambered and sometimes straight. The usual
camber given to weather decks is \ inch per foot of length of beam.
Lloyd's Rules allow the beams of weather decks to be fitted without camber
or with less than the usual amount, in the case of large vessels (30,000
longitudinal number) if at least half the length of the top continuous deck
is covered by erections. On the principle of the arch, it might be thought
that camber should give additional strength to beams, but, as has been pointed

BEAM SECTIONS. io 7

out,* the sides of a ship are not really abutments, so that this can scarcely
be the case.

BEAM SECTIONS.— Beams are fitted of different sections according to
the strength required; most of these being included in fig. 99. Sections
A and B, and in large vessels, section E, are adopted when the beams are
at every frame. G is not often used for ordinary beams ; it is a common
section, however, for special beams built to carry the ship's boats. D and
H are used when beams are to be fitted to alternate frames. Sections F
and J are only fitted where extra strength is required. In way of the
machinery, the material binding the sides of the vessel together is usually
very much cut away owing to the necessity of providing ample space for
shipping and unshipping the engines and boilers ; it is, therefore, of import-
ance to make any beams that may be got across the vessel in that
locality as strong as possible; beams similar to those just mentioned are
usually fitted, with satisfactory results. (? is a form of beam found suitable
for the ends of hatchways, the angle bar being, of course, fitted away from
the hatch opening. In general, where heavy permanent deck weights are
carried, specially strong beams are needed, and the section adopted will be
that dictated by the experience of the designer.

Fig. 99.

In order to obtain special strength, and to allow of substantial knee
connections between the beam ends and the frames, ship beams are not
reduced in depth towards their extremities, as might be done in the case
of an ordinary loaded girder supported at the ends. The reasons for having
great strength at this part of a ship have already been explained.

BEAM KNEES, — We shall now describe a few methods of forming and
fitting beam knees. Several examples taken from Lloyd's Rules, of a common,
and very efficient one when the workmanship is good, is shown at fig. 100.
It is seen to consist of a triangular plate, fitted into the angle between the
top of beam and the ship's side, and well riveted to the beam end and the
frame. Sometimes for lightness, and to minimise obstruction to stowage, the
inner edge of the knee-plate is hollowed. This knee can be fitted to any
of the beam sections given above.

Another way of forming a beam knee is to cut away the lower bulb for a
short distance from the end of the beam, and weld in a piece of plate or bulb
plate, the knee being afterwards trimmed to the size and shape required. This
is called a slabbed knee (see fig. 101). Unless great care is exercised, the

* See Practical Shipbuilding, by A. Campbell Holms.

ioS

SHIP CONSTRUCTION AND CALCULATIONS.

welds of these knees will give trouble ; for this reason this style is not so
popular as the bracket knee, nor as the one we are about to describe. In

Fig. 100.

BEAMS AT EVERY FRAME

BEAMS AT ALTERNATE FRAMES

C. MUST NOT BE LESS THAN Six TIMES
OlAMETLR OF RIVETS

Fig. 101

Fig. 102.

this last case, each end of the beam is split horizontally at about the middle
of th^ depth, as indicated in fig. 102, and the lower part is turned down;

BEAM KNEES. I09

a piece of plate is welded into the space so formed, and, finally, the beam
is cut to shape and size. This knee also depends on the quality of the
welds, but it is stronger than the previous one, and has a fine appearance ;
it is known as a turned knee.

As the stresses are mainly met by the shearing strength of the rivets,
these must be sufficient in number and diameter. Lloyd's Rules require that
in knees under 17 inches deep there shall be not less than 4 rivets of f-
inch diameter in each arm, while knees 40 inches deep require nine f-inch
diameter rivets in each arm ; the number varies between these limits for
knees of intermediate depths.

Obviously, only about half the number of rivets required in bracket knees
will be needed for welded knees of the same depth.

The depth and thickness of a beam knee varies with the depth of the
beam, and the position of the latter in the ship. Generally, beams which
form the top of a hold space are required of maximum depth, the distorting
stresses being greatest at these places. Hence, in steamers where there is but
a single tier of beams, the beam knees are of greater depth than if there were
intermediate decks. The upper deck beam knees in vessels which have a range
of wide-spaced beams below the upper deck, are to be of the scantlings of
the knees of an upper deck tier where it is the only one. In sailing ships,
beam knees at all tiers of beams are to be the same as for upper deck
beams *of similar scantling in steamers having one tier only. It may be
mentioned that Lloyd's Rules require beams in sailing ships to be heavier
than those of the same lengths in steamers having one tier only. These
requirements are very necessary as sailing vessels have no watertight bulk-
heads except one fitted at the extreme forward end ; they are, therefore,
without the rigid transverse stiffening which every steamer possesses in virtue
of her bulkheads, and need the bracing given by beams of special strength and
depth of knees.

All beam knees should measure across the throats at least ^ of the full
depth of the knee.

In large single-deck vessels the beams at every frame are to have plate
bracket knees varying in size with the moulded depth. Thus, in vessels 23
feet and under 24 feet depth, the knees are to be 33 inches x 33 inches ;
and in vessels 26 feet and under 27 feet, 42 inches x 42 inches; the knees
for vessels of intermediate depths varying between these. Deepening the knees
strengthens the frames by shortening the unsupported length ; it also stiffens
the vessel at the deck corners and arrests any tendency to change of form
that might develop when the vessel is labouring among waves at sea.

BALLAST TANKS. — Nearly all modern cargo steamers are constructed to
load water-ballast when necessary, the water being carried in double bottoms,
in peak tanks, in deep tanks, or in some other space specially devised for
the purpose. Frequent^, all these methods are employed together in a single
steamer, when it is desired to be able to proceed to sea without using supple-
mentary stone or sand ballast.

SHIP CONSTRUCTION AND CALCULATIONS.

Ballast tanks are not usually fitted in sailing ships, as, unlike steamers,
they have long voyages to perform and load and discharge comparatively
seldom. In their case, therefore, rapid means of ballasting are of little use.
Another important reason for omitting ballast tanks" in sailing ships is the
saving in first cost. Still, where there have been special reasons for so doing,
double bottoms, and even deep tanks, have been installed in sailing ships.

When water began to be introduced as a means of ballasting, shipbuilders
devised many more or less successful plans, to which we need not here refer, for
economically carrying it, before they arrived at the efficient system now generally
adopted. From the first it was seen that the broken space between the shell-
plating and the tops of the floors in the bottom of the ship was admirably
suited for this purpose, since use could thus be made of space not otherwise
available for profitable employment. In the earliest vessels the tanks were usually
only in one or two holds, and to obtain an adequate ballast capacity the tanks

Fig. 103.

° ° onW

BRACKET

^i o o o o o

o ■ o o 1

1 FLOOR

W.T. COLLARS

^ v

. FRAME CUT

had to be deepened, which had the manifest drawback of encroaching consider-
ably on the cargo space. The advantages, however, of the convenient ballast
were considered sufficient to outweigh this loss, and many steamers were thus
built. The usual plan followed, was to fit longitudinal girders spaced about
3 feet apart on top of the ordinary floors and to cover them with plating, the
tanks being sealed at the sides by carrying the tank top-plating down to the
shell and connecting it thereto by means of an angle bar caulked watertight. It
was at first found rather difficult to make a satisfactory joint at the ship's side.
One method, shown in fig. 103, was to sever the reversed frames in way of the
tank, to cut a hole in the margin plate for the frame so that the former
could go close against the shell-plating, and then to joggle a bar round the
frames and against the shell. The cutting of the reversed frames was compen-
sated for by doubling the frame in the neighbourhood of the tank margin.

Another method, which did not call for the cutting of the reversed frame,
consisted in working a smithwrought angle bar round frames and reversed frames,
and against the shell-plating, and filling in the little apertures left behind the

BALLAST TANKS.

Ill

reversed frames with plugs of wrought iron, the latter being tightly wedged into
place a-nd carefully caulked. Neither of these methods was found to be
very satisfactory, as the abrupt termination of the tanks gave rise to decided
weakness in the structure at the bilges ; moreover, even with careful workman-
ship, watertightness at the margin was difficult to secure.

Both of these objections were eventually overcome by severing the main
and reversed frames at the tank margin, and fitting a continuous bar directly
on to the shell-plating, the loss of transverse strength being made good by
fitting substantial brackets from the frame bar on to the margin plate of the
tank top. This arrangement, known as the M'Intyre System from the name
of its introducer, is, in principle, the one now adopted at the margin in all
vessels having a ballast tank extending over the greater part of the length. In
the earliest vessels built, on this system, the angle connecting the margin-plate
to the shell-plating was fitted inside the tank and the margin-plate connected
to the tank-top by an angle bar ; now, the margin-plate is flanged at the

Fig. 104.

FRAMED REV. FRAME CUT
CONTINUOUS W.T.ANGIE

top and the shell bar brought outside the tank, improvements which have led
to much better workmanship. Fig. 104 shows the improved M'Intyre System.

In constructing a ballast tank extending over a portion only of the length,
a point of importance is the maintenance of the longitudinal strength at the
breaks. To stop the tank structure abruptly at any point would accentuate the
weakness of sections lying immediately beyond. In such cases the usual plan
is to continue the keelsons of the part of the structure clear of the double
bottom, so as to scarph the latter for some distance (Lloyd's Rules require
a minimum scarph of three frame spaces), and to connect them to the
longitudinal girders where practicable.

It very soon came to be recognised that by making a ballast tank con-
tinuous, and for the full length of a vessel, other advantages besides the import-
ant one of carrying water-ballast could be secured. It was seen, for instance,
that the double skin afforded by the tank top-plating would greatly increase a
vessel's safety against foundering, in the event of grounding on a rocky bottom ;
also that the material required for the construction of the tank, being at a con-

112 SHIP CONSTRUCTION AND CALCULATIONS.

siderable distance from the neutral axis, would be very efficient in resisting
longitudinal structural stresses. These considerations led to the adoption of a
continuous ballast tank in many vessels, and when later, the Board of Trade
consented to measure the depth for tonnage in such cases to the inner bottom
plating, a fore-and-aft double bottom became the rule in cargo steamers.

The fitting of a full length tank brought immediate changes in the internal
framing of this part of the hull. It was now found possible to reduce the
depth of the tank as compared with that of one extending over a part only of
the length ; but this made it impracticable to follow the usual plan in building
of fitting longitudinal girders on top of ordinary floors, and a Cellular System
of construction was introduced, and came to be generally adopted.

There are two principal methods of constructing a double bottom on this
system, illustrated in figs. 105* and 106* respectively. The first consists of
longitudinal girders suitably spaced, and floorplates fitted at alternate frames,
the girders being connected to the inner and outer bottoms by angle bars.
By Lloyd's Rules at least one longitudinal girder is required in vessels under
34 feet breadth, whose breadth of tank amidships is under 28 feet, and two
in vessels between 34 feet and 50 feet in breadth, whose breadth of tank
amidships is between 28 and 36 feet. Sometimes the parts are flanged in
lieu of angles, but although the cost is thus somewhat reduced, there being
fewer parts to fit and less riveting, there is a loss in rigidity, for which
reason flanged work here is not very common. In way of the engine space,
owing to the great vibration there due to the working of the machinery, the
floorplates are fitted at every frame and stiffened at their upper edges by
double reversed bars ; floorplates must also occur at the boiler bearers. As a
rule flanged work is not resorted to in this region.! Before and abaft the
engine space, at those frames to which no floorplates are attached, brackets,
which in medium-sized! vessels should be wide enougn at the head to take
three rivets in the vertical flange of the intermediate reversed angles for 4
the vessel's length amidships, are fitted to the centre girder and margin
plate, binding these parts together and strengthening them to resist the stresses
set up by the action of the water ballast when the vessel is in motion among
waves. The reversed bars in way of these intermediate frames are riveted to
the tank top-plating, to Avhich they act as stiffeners ; frequently, however, they
are dispensed with and the inner bottom slightly increased in thickness in lieu.

The side girders and floors are pierced with manholes to give ready access
to all parts of the tank, a considerable saving in weight being also thus
effected. The centre girder is more important than the others, forming as it
does a kind of internal keel ; it has, therefore, heavier scantlings, is not re-
duced by manholes except perhaps at the extreme ends, and is stiffened top
and bottom by heavy double bars (see fig. 105). In the earliest vessels built
on this plan, the longitudinals were continuous and the floors intercostal, the

* Figures taken from Lloyd's Rules.

t See Remarks on Stiffening of Double Bottom at Fore End, p. 116.

X When the longitudinal number is 20,000 and above.

BALLAST TANKS.

113

former, owing to their distances from the neutral axis, being considerable
elements in the longitudinal strength. Nowadays, it is usual for the floors
to be continuous and the girders intercostal, an arrangement leading to greater
simplicity of construction — a most important point— and to some reduction in
longitudinal strength, which, howe'ver, in modern vessels, calculations show to
be still ample. An important advantage of the latter arrangement is a great
increase in the stiffness of the bottom, the comparatively short floorplates
obviously having greater strength and rigidity than long fore-and-aft girders.

Fig. 105.

SECTION AT INTERMEDIATE FRAME

£23" *

£=H*

/*■— 1 1 ft

PLAN

-H-,.-^^.

BRACKET

CENTRE GWOtR

^^Tf F""^

SIDE GIRDERS

ti or

When longitudinals are continuous, in order that their efficiency as strength
elements may not be impaired, the manholes through them should be as few
as possible, and those in different girders shifted well clear of each other
transversely.

When the rule lengths of vessels exceed 400 feet, and in single-deck
vessels which exceed 26 feet moulded depth, the above plan of framing the
inner bottom is not considered adequate, and the second method, shown in
fig. 106, should be adopted. In this case, the floorplates are at every frame

ii 4

SHIP CONSTRUCTION AND CALCULATIONS.

and continuous from centre girder to margin plate on each side, while the
longitudinals, except the centre one, are intercostal. The floorplates and side
girders have lightening holes, one or two through the floors into each cellular
space, and one through every intercostal girder plate. Fewer side girders are
required by this plan, only a single one being necessary on each side of the
centre if the ballast tank be under 36 feet in width and the breadth of
ship under 50 feet, and only two if the tank be under 48 feet and the
ship under 62 feet in breadth, the number being proportionately increased in
larger vessels ; the spacing of the girders with the closer floors gives approxi-
mately the same extent of unsupported area of shell and tank top-plating as
in the previous case. The intercostals are attached to the floors and to the

Fig, 106.

FLOOR AT EVERY -FRAME'

PLAN

inner and outer skins by riveted angle bars or flanges ; and the floors, as well
as being riveted to the frames and to angle bars under the inner bottom plating,
have angle connections to the centre girder and the margin-plate; the centre girder
attachments, consisting of double angles for half length amidships when the trans-
verse number reaches or exceeds 66. In the latest vessels of this size, single
angle attachments between the floors and centre girder have sometimes been
adopted, the flanges being double riveted. It is seen that as regards the in-
ternal framing of the inner bottom, the longitudinal strength in this last plan is
somewhat less than in the previous one, but in view of the tendency towards
increase of breadth in modern vessels, demanding considerable transverse
strength, and of the greatly enhanced stiffness of the bottom on account of
the numerous deep floorplates, it would appear that the continuous floor on
every frame method of construction is the better one, particularly as the

BALLAST TANKS.

absence of the bracket work required at the intermediate spaces in the previous
plan renders the work of simpler construction. This arrangement is at anyrate
a favourite with many builders, who have frequently adopted it in much
smaller vessels than those requiring it owing to their size. Lloyd's Rules
recognise the greater strength of this plan over the previous one by allowing
the shell-plating (except the flat keel and garboard strakes) in way of the
tank, when -52 to '66 of an inch in thickness, to be slightly reduced; when
the plating exceeds '66 of an inch in thickness, no reduction is allowed.

Fig. 107.

TANK. SIDE BRACKET

PLAN

°1

L_.

■ 1

1 -

] lol a

ANCLE GUSSET

DOUBLE ANCLES

MARGIN PLATE

A part of the structure at which weakness has often been found developed
in vessels fitted with double bottoms is where the side framing is attached
to the margin plates. Experience with actual vessels has shown the need of
making this connection amply strong, many of them having exhibited signs of
movement in the shape of loose rivets in the angles connecting the side
brackets to the margin plates. In Lloyd's Rules a minimum breadth of
margin plate is given, with a corresponding minimum number and size of rivets
for making the bracket connection. In small vessels, single angles are con-
sidered sufficient to join the side brackets to the margin-plate, but with

Il6 SHIP CONSTRUCTION AND CALCULATIONS.

increase in breadth and depth, particularly the latter, double bars or equivalent
single bars with double-riveted flanges quickly become necessary for this pur-
pose, over some portion at least of the vessel's length. Experience has shown
the fore end to require special attention in this respect, and Lloyd's Rules
demand double angles from the collision bulkhead to one-fourth the vessel's
length from the stem in vessels of moderate size. Besides the foregoing,
with the growth of vessels additional strength becomes necessary at the
tank margin, which is provided by fitting gusset-plates to the tops of the
wing brackets and to the sides of the tank top-plating. In recent instances,
angles have been substituted for the gusset-plates with good results. Detailed
sketches of these arrangements are shown in figs. 105, 106, and 107. These
gusset-plates or angles are fitted at every fifth, fourth, third, second or single
Irame, according to the vessel's size, the limits being fixed in each case by
the transverse number and the value of d, i.e., the length of unsupported
frame.

The fitting of the inner bottom plating calls for little comment. Lloyd's
Rules recommend that it be arranged in longitudinal strakes and the butts
shifted well clear of each other and of those of the longitndinal girders, when
these are continuous, and this is usually done. In some districts, notably the
N.E. coast, transverse strips have been fitted under the watertight bulkheads to
allow the building of the latter to be proceeded with at an early stage of the
work, but the system cannot be otherwise commended. To save the fitting of
packing pieces, inner bottom plating is sometimes joggled at the seams, but
objection has been raised to the depression thus caused in the surface, as
forming lodgments for water, particularly where ceiling is laid, rapid corrosion
resulting in consequence. Like other longitudinal material, the thickness of
this plating is reduced at the ends; it is increased in way of the machinery
space to give additional strength, but more particularly to allow for <the
corrosion which takes place there.

Structural efficiency in double bottoms would not be obtained were care
not bestowed on the riveted connections. It is especially important that the
centre girder should not be weakened at the butts ; except in the case of
small vessels, therefore, these must not be less than treble riveted amidships,
and in very large vessels they should be quadruple riveted.* The butts of
side girders, where continuous, should be double riveted, and in laro-e vessels
treble riveted. The tank top-plating is an important element in both the
longitudinal and transverse strength, and the riveting of butts and seams calls
for careful consideration. The butts of the middle line strake, and those of
the margin plate, must be at least double riveted throughout ; in large vessels
they should be treble and in the largest vessels quadruple riveted. The re-
maining butts of inner bottom plating are to be double riveted for half

* In Lloyd's Rules, and also in those of the British Corporation, the requirements as to
riveting of butts and edge joints at any part of a ship are fixed by the thickness of the
plating and the position of the part.

PEAK TANKS. 1 17

length, and in large vessels treble riveted. The edges of the middle line
strake, where the transverse bending stresses are greatest, except in small
vessels, should be double riveted, and in the largest vessels this should apply
to the remainder of the plating; medium-sized vessels have single riveted edges
clear of the middle line strake. All the preceding connections are usually
overlapped.

On page 73 reference was made to special stresses which come upon the
fore-ends of full vessels when sailing in light trim among waves at sea. To
resist these stresses, such vessels are provided with extra strengthening forward
in way of the inner bottom. If built on the cellular system with compara-
tively closely spaced longitudinals, continuous or otherwise, and floors at alter-
nate frames, the floors are to be fitted to every frame, and the main frames
doubled from margin plate to margin plate from the collision bulkhead to
a fifth of the vessel's length aft, measuring from the stem ; also the three
strakes of plating next the keel are to maintain their midship thickness for-
ward to the collision bulkhead. If built with continuous floors at every
frame, the frames are to be doubled, and the shell-plating increased, as in
the last case ; but in addition, special intercostal girders must be fitted of
a depth equal to half that of the centre girder, and be extended as far for-
ward as practicable. In both cases the rivets through the plating and frames
in this region are to be of closer spacing than elsewhere. It should be
mentioned that in vessels having ballast tanks constructed with longitudinal
girders on top of ordinary floors, and in those without inner bottoms, if of
full forms, adequate strengthening of a similar nature to the above is necessary.

PEAK TANKS, — In steamers, after-peaks are now usually adapted for water
ballast ; in some cases, the fore-peak is also so constructed, but more rarely.
The principal value of peak tanks is, of course, their trimming effect ; they are
like weights situated at the extreme ends of a lever poised at the middle, and
have great power in this respect. In strengthening these compartments for
their work, we have to bear in mind the special nature of the stresses set
up by the load. Unlike ordinary cargo it does not lie on the floors, but
presses immediately on to the shell, thus inducing severe stresses on the frame
rivets which bind the shell-plating to the structure ; for this reason the pitch
of such rivets is not to exceed 5 diameters. It is not usual to increase the
shell-plating or framing in thickness, as owing to the shape at this part these
are amply strong. The boundary formed by the hold bulkhead, however,
requires special attention ; being flat and of considerable area, care has to be
taken to prevent bulging ; this is done by thickening the lower plating if the
tank is a deep one, and making the stiffeners heavier and closer spaced than at
ordinary bulkheads. A centre line bulkhead or washplate is required in all
such compartments, to prevent the water from damaging the structure by
dashing from side to side in the event of a free surface, and to minimise
the effect which such free surface would have on a vessel's stability. Such
washplates need not be of strong construction, but should be securely attached
to the bulkhead and underside of the deck.

SHIP CONSTRUCTION AND CALCULATIONS.

DEEP TANKS.— These usually consist of ordinary hold compartments
specially strengthened to carry water ballast; one or two of them are fre-
quently fitted in large modern cargo steamers. Where one only is required,
the compartment immediately abaft the engines is usually adapted for the
purpose ; where there are two, they are generally placed one at either end
of the engines and boiler space. When situated thus, the machinery being,

Fig. 108.

ELEVATION

BRACKETS -

SECTION

i c 5 ° ' -j Tj/ PECK (TVT

PLAN.

ALTERNATIVE PLAN, FRAMES CONTINUOUS

ELEVATION

SECTION

of course, assumed amidships, the water ballast has its greatest power in
sinking the vessel bodily, without materially changing the trim.

The same system of strengthening is followed here as in the peak
tanks, but deep tanks being larger, the stiffening has to be correspondingly
increased. The end bulkheads must have heavy vertical and horizontal
stiffeners of bulb angle, or other section, fitted on opposite sides and
bracketed at their ends, in the one case to the double bottom and deck,

SPECIAL BALLAST TANKS.

II 9

and in the other, to the ship's sides. A centre line division is required,
and must be of strong construction, as it takes the place of pillars, as well
as acting as a wash plate. It should be connected by double angles to the
deck and double bottom, and have substantial, close-pitched stiffeners bracketed
at top and bottom. Cases have occurred in which the action of the water
has swept this bulkhead entirely from its boundary connections, showing the
need of having the latter specially strong. The deck forming the top of the
tank is required to have beams spaced on every frame, and there should be
large beam knee connections to the sides, as severe strains have been found
developed at this part due to the action of the water in a partially-filled tank,
when the vessel has been in motion at sea. Midway between the centre line
and the ship's side, runners are required under the deck beams, and, in order

PLAN

to tie the top and bottom of the tank together, against the lifting forces
exerted by the contained water when the vessel is in motion among wave^ a
row of strong pillars are required. The riveting through the frame and shell-
plating is of close pitch in way of deep tanks, as the load acts directly on
the shell-plating. At the ship's side watertightness is secured in much the
same way as at the margin of a double bottom. A continuous bar is fitted
and caulked to tank top plating and shell, the side frames, which have, of
course, to be severed, being bracketed to the tank top (see fig. 108); some-
times, where the side framing consists of frame and reversed frame, only the
reversed frames are cut, the frames being doubled in the vicinity and water-
tight collars fitted ; in this last case, the severing of the reversed frames is
further compensated for by fitting brackets at alternate frames (see fig. 109).

120 SHIP CONSTRUCTION AND CALCULATION^.

Access to deep tanks for the purpose of shipping cargo is obtained by means
of watertight hatchways, one of which, owing to the centre division, is required
on each side of the centre line.

OTHER TANKS. — As well as the foregoing, in many steamers special
arrangements have been devised for carrying water ballast. Thus, in the
Harroway and Dixon type of vessel, corner spaces under the deck are cut
off from the holds, and specially strengthened for this purpose (see fig. 82).
In the Burrell type the bilge corners are built up and utilised for water
ballast (see fig. 84). In the latest Ropner Trunk type, portions of the trunk
space have been specially strengthened to the same end. The M'Glashan
system, which consists in continuing the double bottom up the sides of the
ship to the height of the deck, is also worthy of mention. The chief point
in favour of these arrangements is the high position thus secured for the
centre of gravity of the water ballast, leading to steadiness and general good
behaviour when in a seaway.

TESTING OF TANKS.— On completion, ballast tanks are tested for
watertightness by putting them under water pressure. Each compartment of
a double bottom intended for water ballast is pressed by a head of water
to the height of the load waterline as being the greatest pressure it need
bear in actual service. Peak tanks and deep tanks are tested by a head
of water 8 feet above the top of tank, but the head must in no case be
less than to the height of the load waterline.

PILLARS. — The importance of pillars in a ship structure has already
been pointed out. It was shown that as struts and ties they communicate
stresses from one part to another, and thus cause the strength of the various
parts of the structure to act together. Short pillars are more effective than
long ones, as the latter are liable to collapse by side bending at a much
less strain than that represented by the compressive strength of the material.
Pillars are, therefore, increased in diameters with their lengths; e.g., in a vessel
of 55 feet beam with two rows of pillars, the latter, if just under 8 feet long
and supporting a beam of a third deck, should have a diameter of 4 inches,
and if just under 22 feet a diameter of 5! inches; intermediate lengths having
diameters between these. The strength of pillars should also advance with
the loads they have to bear; for instance, those in the upper erections, since
they have only the weight of the deck structure and load to support, may be
of comparatively small size ; those fitted under second or third decks below
the upper deck, which may, therefore, have heavy loads of cargo to support,
should be of considerable diameter. In the holds, too, the side pressure of
the cargo is liable to bend the pillars unless they be of substantial diameter.

As well as acting as struts and ties, pillars greatly augment the strength
of the beams they support. A pillar placed below the middle of a rect-
angular beam, supported at its ends but not fixed there, will double the
strength of the latter and greatly increase its rigidity ; beams in ships have
fixed ends, but except as modified by this circumstance, the strength value of
middle line pillars is equally great. If two pillars be fitted to a beam so as

PILLARS.

121

to divide its length equally, the effective span is a third of its original value,
and the strength of the beam is correspondingly increased; and so on for any
number of pillars. Use of this is made in vessels as they increase in breadth ;
for instance, beams 43 feet and under in length require only a centre row of
pillars, but when they exceed this length, two rows become necessary; when
the length of beam exceeds 60 feet, an additional row is required, placed

Fig. 110.

1 u

Fig. 111.

Fig. 112.

Fig. 113.

Lln_

Fig. 114.

Fig. 115.

Fig. 116.

one at the centre line, and another at each quarter breadth of the vessel, those
in the latter rows being hence called quarter pillars. At the ends of the
vessel, as the beam decreases in length, the number of rows of pillars may
be correspondingly reduced. Where there are several decks, the various rows
of pillars should be arranged as nearly as possible over one another, in order
to rigidly join the upper and lower parts of the ship's structure.

HEADS AND HEELS OF PILLARS.— The forms of the heads and
heels of pillars are governed by the nature of the part of the vessel to

122

SHIP CONSTRUCTION AND CALCULATIONS.

which connection is made. Figs, no and in illustrate two methods of
attachment to bulb-tee beams ; the first is the usual one, the second is not so
common but has the advantage of gripping the beam round the bulb, and
so relieves the rivets when the pillar is under tension. Figs. 112 and 113
show the connection to H and channel beams respectively. When beams
are fitted on every frame, the pillars being at alternate frames, it is necessary
to have runners under them in way of the pillars so as to support the

Fig. 117.

^^^p—^^vt — ^ ^tm

Fig. 118.

Fig. 119.

Fig. 120.

Fig. 121.

_V M

Fig. 122.

Fig. 123.

intermediate beams. These runners should consist of double angles, but may
be of other approved form. Figs. 114 and 1 15 show two styles of the
beam runner ; it will be observed that the attachment to the beams is by
means of a riveted angle lug; where the beams are of channel section this
is unnecessary. Fig. 116 is a suitable form where the pillars have to be
reeled for shifting boards, as is frequently the case with those at the middle
line in cargo vessels. The plan adopted is to fit consecutive pillars on

HEADS AND HEELS OF PILLARS.

123

opposite flanges of the channel runner, thus forming two lines between which
the shifting boards may be reeved. Where intercostals to the deck-plating
are required, as in the case of quarter pillars to deep tanks, or where pillars
are widely spaced, they are fitted in various ways, figs. 117, 118, and 119,
also figs. 126 and 127, showing some of these with pillar head attachments.
The intercostals transform the beam runners from simple ties into strong girders
eminently qualified to stiffen the deck and to distribute the stresses.

The heel attachments of pillars are not so varied in form as those of the
heads. A common one is shown in fig. 120, and is seen to consist of a
horizontal shoe forged on to the lower end and through riveted to the deck-
plating or to the beam, as the case may be. A favourite method when

Fig. 124.

: o o :

<&?

SHOE- BEVELLED

s^&>

72117/ /T7VSf?S7K EZZZZZ

ALTERNATIVE PLAN

Z &AVY ZZSzdssfea SSaZSaSS ^-w-wwy;^

/ ' : i

the heel comes on an inner bottom, is to rivet a short bulb angle or tee
lug to the plating, and connect the pillar to the vertical flange (see figs.
121 and 122). This plan is sometimes followed for making attachments to
steel decks when the latter are to be wood sheathed, the vertical flanges of
the lags being made deep enough to allow of the heel of the pillar being
above the wood deck (see fig. 123) to facilitate the caulking of the latter
at this part.

The end attachments of pillars should consist of at least two f-inch
rivets. When they reach a length of 18 feet or a diameter of 4 inches there
should be three rivets in each end. Pillars 5 inches or above in diameter
require a four-rivet connection at the ends.

124

SHIP CONSTRUCTION AND CALCULATIONS.

Pillars are frequently required to be portable ; in way of the hatches,
for example, they must be removed when loading many kinds of cargo.
Bolt and nut fastenings are often adopted (see alternative plan in fig. 124), but
occasionally, particularly at the heels, these are found impracticable or un-
desirable, and arrangements such as those of figs. 124 and 125 are resorted
to. When fitted in either of these ways a pillar can, of course, only act as
a strut.

WIDE-SPACED PILLARS. —While several rows of close-ranged pillars
are valuable to a vessel as regards her strength, from a point of view of
stowage they are obviously somewhat of a drawback. With almost all
cargoes, the pillars, particularly those in the wings, must give rise to a con-
siderable amount of broken stowage, and although by splitting up the cargo
they prevent damage through side pressure when the vessel is rolling, many

Fig. 125.

WOOD DECK

owners have sought to dispense with them where possible. For example,
in vessels of a breadth requiring say, three complete rows of pillars, it is
permitted, and, for the above reasons, usually preferred, to substitute instead
one complete middle line row, with two rows in the wings at wider spacing.
With this modified arrangement, however, beam runners, having intercostal
attachments at each deck, must be fitted in way of each line of quarter
pillars, the scantlings of the intercostals and pillars being governed by the
spacing of the latter, the breadth of deck to be supported, and the probable
load. In Lloyd's Rules, Tables are provided giving the scantlings of wide-
spaced pillars and of the girders at their heads.

For many trades, as mentioned in the previous chapter, even when thus
spaced, the pillars have been found to be too numerous, and the centre row
has been dispensed with, and a very wide spacing aqlopted for the quarter

WIDE-SPACED PILLARS.

125

pillars, in some vessels not more than two aside being fitted even in long
holds. In such cases the decks have been supported, and the loads com-
municated to the pillars, by means of runner girders of enhanced strength,
and the greater stresses brought upon the pillars have been met by making the
latter of special size and construction. Figs. 126 and 127 illustrate two
arrangements to Lloyd's requirements. These pillars, it will be noted, are

Fig. 126.

/BEAMS
HANMELS

GIRDER IO*3V3V-60
DOUBLE CHANNELS

PILLARS 12 DIA*
PLATING 54"

TANK TOP

— FLOOR
INTERCOSTAL GIRDER

stepped on the tank-top at the junction of a floorplate and intercostal girder.
This is necessary for rigidity, and when pillars cannot be so placed they
must be similarly supported by means of brackets or have seatings built on
the tank-top.

OUTER BOTTOM.— The most important part of any ship is the outside
shell-plating. Its leading function is to give the structure a capacity to dis-
place water, but, besides this, being spread like a garment over all the inner

126

SHIP CONSTRUCTION AND CALCULATIONS.

framing and securely riveted thereto in every direction, it binds the whole
together and enables the various parts to efficiently resist the severe stresses
brought upon them when the vessel is in lively motion among waves at sea.
Every part of the shell-plating is of importance, but owing to their positions
some parts must be of greater comparative strength than others. We have
seen that the greatest longitudinal bending stresses come upon the upper
and lower works, and the least in the vicinity of the neutral axis; so that,

Fig. 127.

with the vessel upright, the sheerstrake at the top, and the keel, garboard,
and adjacent strakes at the bottom, are most severely stressed, while the

material at about mid-height — the position of the neutral axis is stressed

least. It has also been pointed out that the above conditions become modi-
fied when, through the rolling of the vessel, the side-plating is raised towards the
top of the girder away from the neutral axis and has to sustain a much
increased stress; this, and the fact that the longitudinal sheering stresses

OUTER BOTTOM. 1 27

where they occur in the length, are a maximum at the neutral axis, must
be borne in mind when apportioning the scantlings to the various parts of
the shell-plating. Towards the ends of the vessel the structural stresses
are less than amidships, and the thicknesses are reduced; this applies to all
longitudinal materials in a ship.

The Rules of all classification societies require the sheerstrake, the keel-
strake, and those adjoining, to be specially heavy, the strakes from above the
upper turn of the bilge to the sheerstrake being of smaller scantlings. Of
course, in certain places, where severe local stresses may be anticipated,
special strength is introduced. Thus, the afterhoods of the strakes which
come on the sternposts in steamers are retained of midship thickness to
withstand the stresses which the working of the propeller brings upon that
part of the structure. For the same reason the plates in the immediate
vicinity of the propeller shaft, called boss plates, are increased in thickness
beyond that required for the same strakes midships. Usually, the shell-plating
is thickened forward where it has to take the chafe of the anchors ; and
in some special vessels the plating in the vicinity of the stem is thickened
to withstand ice pressure.

The actual thickness of the various parts of the shell-plating of a vessel
are governed by the size of the latter. For example, in a small vessel,
say one 90 feet or 100 feet in length and under 10 depths to length, with
a longitudinal number under 3350, the shell scantlings in fractions of an
inch would be : — keel-plate, amidships '44, ends "36 ; garboard strakes, where
there is a bar keel, amidships '34, ends '30 ; shell-plating, from flat keel-
plate or garboard strake, to strake below sheerstrake, amidships '30, ends '26;
sheerstrake '32 ; strake below '3, ends '26. In a cargo steamer of average size,
say about 360 feet long, with a proportion of length to depth of between
11 and 12, and a longitudinal number of 28,400, the corresponding scant-
lings would be : — keel-plate, without doubling, '94 to "66 ; garboard strake
with a bar keel, '64 to '54 ; from flat keel-plate or garboard strake to
upper turn of bilge, "6o to "46 ; from upper turn of bilge to strake below
sheerstrake, *6o to '44; sheerstrake, 72 to '44; strake below sheerstrake,
•62 to '44. In each case, the second thickness is that at the ends. In
each of these sets of scantlings, if the keel strake and sheerstrake be
omitted, there is a comparative uniformity throughout ; this is what might
have been expected from our considerations above. Another point of interest
is the small amount of taper towards the ends in the scantlings of the
small vessels, compared with those of the other. Structurally, the end
thicknesses in the smaller vessel are probably too great, but as even the
maximum thicknesses are small, the necessity of allowing for wear and tear
prevents the liberal reduction permissible in the heavier material of the
larger vessel.

Having decided upon the scantlings, the next point of importance is to
arrange the end joints of the plates forming the various strakes. These
should be disposed in such a way as to avoid having too many weak points

128

SHIP CONSTRUCTION AND CALCULATIONS.

In the same transverse section. Lloyd's Rules stipulate that joints in ad-
joining strakes must not be nearer to each other than two spaces of frames,
and those in alternate strakes at least one space clear. They also demand
that the end joints or butts of the sheerstrake be shifted clear of those of
the deck stringers by two frame spaces, and the end joints of the garboard

Fig. 128.

i i

i i i i • i '
' ! i

Ui iii

i :! i

! i I ' ' * ' ' I ' I ! t
l r ,-l 1t j- , » ; I;,; ^-U-fr

■ i Hi^J

^iirn

1 1

1 :

: . ! ii 1 ■ ■ : : ii: I ' I i ■

strake on one side of the ship clear by a like distance of those of the same
strake on the other side. This latter precaution is of course because of the
proximity of the garboard strakes, only the keel separating them. Fig. 128
illustrates a shift of joints or butts embodying the minimum requirements of
Lloyd's Rules. It will be observed that there are here four passing strakes

Fig. 129.

tt~i

I 1 •

. ; i 1 j ii

1 — r 1 '

nU«Uj

I I

I " » - .

I ' l!

T ri ~1

' 1 1

1 ■ <

Tfry^

1 if , I I I

4--t.i l -- L 4-fr.»4

rt'^ , ^^I't

I ' 1 ' 1 I I 1 it » I I

1 ' I I 1 I l ' I 1 1 ! t •

ihTTi j ' » . • 1 1 1 * ,

■■^m ......

1 l <

++M4±

4+r4

lj44JUj3

hrnrrn

r^-

^^tiUJr

iLUWJJiUiJULU'JiilL

between joints which occur in the same frame space. Nowadays, as plates
may be rolled to almost any desired length, a better disposition of joints
than that of fig. 128 is easily obtainable. It is not, however, necessary to
have more than a certain number of passing strakes between consecutive butts
in a frame space, no more, indeed, than is required to ensure the same strength

OUTER BOTTOM.

129

at a line of end joints as at a line of frame rivet holes, the latter being
taken as the standard because the loss of sectional area is there unavoidable.
In strength estimates, of course, allowance must be made for the assistance
rendered to the joints by the edge rivets between the joints and the
frames. By actual calculation the best arrangement in any given case could
be arrived at; such calculations are in practice, however, seldom called for,
as a good constructor from his experience is perfectly qualified to devise an
altogether satisfactory disposition of joints. Figs. 129 and 130 illustrate
different arrangements carried out in cargo vessels recently built.

A point to be noted in arranging shell-strakes is the question of their
breadth. As very broad plates lead to a saving in riveting and in the
work of erecting — fewer plates being required for the ship than where the
strakes are narrow — they are naturally popular with builders. There are ob-
jections to their use, however, in that the lines. of weakness which occur at

Fig. 130.

the butts are increased thereby, and that the edge laps being fewer than
where the strakes are narrow, there is some loss of longitudinal stiffness.
For these reasons excessively wide strakes are not adopted by the best ship-
builders. Obviously, plates may be broader in large than in small vessels,
as there will still be a sufficient number of strakes in the former case to
give a good shift of butts. Lloyd's Rules fix the maximum breadth of
strakes at 48 inches in vessels 20 feet in depth, and at 66 inches in
vessels 28 feet in depth and above.

The methods of forming the joints of the plates at the edges and ends call
for careful attention. In early vessels the edges of the strakes were arranged in
clinker fashion (as in fig. 131), but this had several objections, the principal
one being the need of tapered slips at the frames ; it was, therefore, aban-
doned in favour of the now universal raised and sunken strake system, shown
at fig- 132. An obvious advantage of this style over the preceding one is
that only half the number of frame slips are required, which, being parallel,

i 3 o

SHIP CONSTRUCTION AND CALCULATIONS.

are also less costly and more easily fitted. Other advantages consist m the
increased efficiency of construction consequent on having, at least, half the
plating directly secured to the framing without packing, and in the possibility

Fig. 132.

of fitting all the inside strakes simultaneously instead of one at a time, the
method of plating on the clinker system. In many modern vessels frame slips
or packing pieces have been dispensed with altogether, the shell plates being

OUTER BOTTOM.

131

dished or joggled at the edges (as shown in fig. 133), so as to bring their inner
surfaces directly on to the flanges of the frames. The advantages claimed
are — more efficient riveting, there being two instead of three thicknesses to
join, and less weight and cost in the materials of construction. It has also
been said that there is a saving in displacement, but there is very little in
this, as against the saving in weight of ship at each frame, there is the loss
of displacement due to the depression of the plating between the landings.

Fig. 133.

Fig. 134.

This depression, too, it should be noted, causes a reduction in the internal
capacity for grain cargoes. Moreover, there is no saving in workmanship, as
the joggling of the plates has to be put against the fitting of the packing.
On the score of appearance alone, many owners object to the system. Its
greatest drawback, however, is probably found in the increased cc*st and diffi-
culty of carrying out repairs to the sherl-plating, when, through the accidents
of collision or grounding, these become necessary.

132

SHIP CONSTRUCTION AND CALCULATIONS.

The fitting of packing pieces may also be obviated by joggling the frames
(fig. 134). As the plates are not dished, there is a saving in displacement
represented by the weight of the packing pieces; also, there is no loss in in-

Fig. 135.

ternal capacity. In the case of repairs, if conveniences for joggling be not avail-
able, renewed frames may be put in without joggling, packing being used in the
ordinary way. For these reasons, this plan has found favour with many owners.

OUTER BOTTOM. 133

In some yachts and other special vessels, instead of overlapping, the
edges are butted, thus necessitating inside strips at the seams (see fig. 135).
By this method double the number of rivets is required ; it also entails a
greater weight of material and is considerably more costly than the common
method. The flush joint has a decidedly good appearance, but obviously the
important considerations of cost ancf weight are sufficient to debar its use in
any but the vessels above referred to.

The number of rows of rivets required in the longitudinal seams is
governed by the thickness of the plating, and, therefore, by the size of the
vessel. In small craft, in which the shell-plating is less than -36 of an inch
in thickness from the keel-plate to the strake below the sheerstrake, a single
row of rivets is sufficient; in larger vessels, in which the plating in the same
region is '46 of an inch, or more, a double row of rivets is required in the
seams. The landing edge of the sheerstrake, on account of its importance,
should always be at least double riveted.

Until recently, double-riveted seams were considered sufficient even for the
largest vessels, but for reasons already given {see page 69), in vessels of 480 feet
and upwards, built to Lloyd's requirements, or where the thickness of the side-
plating is less than '84 of an inch, it is now necessary to treble rivet the
seams in the fore and after bodies for one-fourth the length and one-third the
depth in the vicinity of the neutral axis. The seam riveting of vessels of
from 450 feet to 480 feet in length, is also to be increased at these parts,
proportionately to their length. In very large vessels which have side-plating
•84 inches in thickness or above, the edges must be treble riveted for i the
length midships. Fig. 136 illustrates single, double and treble riveting at seams.

The end joints of shell-plates may be formed either by butting or over-
lapping ; examples of single, double, and treble riveted joints, formed in both
these ways, are shown in fig. 137. In making the sketches, overlapped-edge
seams on the raised and sunken-plate system have been assumed ; with the
edge seams formed otherwise, there would be some differences in the details
of the end joints.

The question of the number of rivets is decided by the percentage of
strength required in the joint compared with the solid plate. In no vessel,
however, should the end joints of the shell-plating be less than double riveted.
With increase in size of vessels, the need of greater longitudinal strength has
made it essential to resort to treble and quadruple riveting at the end joints.
In the largest vessels, especially when the proportion of length to depth is
excessive, double buttstraps treble riveted are required for the end joints of
the sheerstrake and neighbourhood.

In comparing overlapped joints with those having buttstraps, notable points
in favour of the former are : — reduction in number of rivets, saving in weight
of materials, and reduced cost of construction. It has been objected that
the projections due to overlaps cause a drag on a vessel's speed, on account
of the dead water which they create ; also that the overlapped joint has not
the nice appearance of the flush type with the strap inside ; but the question

r 34

SHIP CONSTRUCTION AND CALCULATIONS.

of cost has, for cargo vessels at anyrate, quite established the supremacy of
the former.

Both lapped joints, and those having single straps, have a tendency to
open when under stress, due to the line of the resultant stress not passing
through the middle of the joint, thus causing a bending action to be

Fig. 137.

•i \

OOOOOOOOl

ooooooooj

•:

developed. Joints having double straps have not this defect, as the resist-
ance to the pull is equal on both sides of the plate.

It may here be mentioned that strained joints situated below the waterline
usually leak, this being the unmistakable sign that undue straining has taken
place. In such cases, recaulking is resorted to, and, although a cure is

OUTER BOTTOM.

i3S

Fig. 138.

SECTION AT AB.
Cat away where dotted in way of edge lap.

Fig, 139.

TZ3

SECTION AT AB.
Dooted part cut away for breadth of edge lap.

136 SHIP CONSTRUCTION AND CALCULATIONS.

generally thus effected with overlapped joints, the same cannot be said for
those of the butted type. In the latter case, if the opening at the seam
is at all wide, an attempt to recaulk it will but make it more unsightly ;
moreover, repeated treatment of this sort renders the material brittle and liable
to break away. A better way to deal with such a case is to fill the seam
with a suitable cement, taking care first to thoroughly clean out the rust ;
this restores the flush appearance and makes the joint watertight.

With overlapped end joints, some difficulty is experienced in obtaining
good work where crossed by the seams of adjoining strakes. Until a few
years ago, the usual method of construction was to resort to packing pieces,
but this caused unfairness in the landings at each lap joint, and, unless
great care were taken in fitting the packing, it could not be satisfactorily
caulked. A method now largely adopted is that shown in figs. 138 and
139. In the first figure, which refers to a joint in an outer strake, the end
of one plate in way of the joint is seen to be tapered away for the breadth
of the edge lap, so as to allow the landing of the outer strake to bear
evenly on the inner one without the necessity of packing. In the case of a
joint in an inner strake a similar plan is followed (fig. 139). An objection
to this scarphing of the seams at the end joints is found in the increased
difficulty of executing repairs, where, as may sometimes be the case, the
requisite machinery may not be available.

In working shell-plates care must be taken to shear from the faying
surfaces — i.e., the surfaces which come together to form a joint — or the rag
left by shearing must be chipped off, otherwise it will be difficult to close
the work. To facilitate caulking, plates forming outside strakes are usually
planed at edges and at one or both ends. In the case of plates forming
insides strakes it is necessary to plane one end only, the other end and the
edges not requiring to be caulked. When finally shaped and punched shell-
plates are secured in place by bolts and nuts which should be sufficient in
number to thoroughly close the plates joined, otherwise difficulty will be
found in obtaining satisfactory riveting.

RIVETS AND RIVETING.— Nowhere will a lack of efficiency more
quickly show itself in the hull of a vessel than at the riveting. The
thicknesses of the materials may be well distributed, and the joints carefully
shifted from one another, but if the riveting be weak, the straining of the
vessel will soon slacken the connections and render her leaky and unseaworthy.

The strength of a riveted joint depends on the aggregate sectional area of the
rivets in it, on the spacing of the latter, centre to centre, on the style of the
heads and points of the rivets used, and on the material and workmanship.

The number of rivets in a joint varies according as the latter is to be
a single, double, treble, or quadruple riveted one — that is, according to the
percentage of strength required in the butt as compared with the unpierced
plate. We have already indicated generally when and where each class of
joint should be employed in a ship, and we now propose to deal with the
details of these riveted connections.

RIVETS AND RIVETING.

137

The sizes of rivets may be said to be governed by the thickness of the
plates they join. If the provision of adequate shearing strength had alone
to be considered, wide-spaced rivets of large diameter might be fitted, but
for watertight work the distance between the rivets next the caulking edge,
especially in thin plates, must not be too great, otherwise the water pressure
will cause a tendency to flexibility in the plate edges between consecutive
rivets ; a comparatively close pitch is also necessary to resist the opening
action of the caulking tool. For the same reason, the line of rivets next
the caulking edge should not be further from the edge of the strake than
about twice the thickness of the plate.

If rivets were of large diameter compared with the plates joined, their
shearing strength would greatly exceed the strength of the material between
them and the edge of the pla'.e, and the connection would consequently
fail, when under stress, by the rivets tearing through the plate edges. To
prevent this happening, a maximum diameter of rivet is fixed at about twice
the thickness of either of the plates joined. With the rivet at one diameter
from the edge of the plate, it can be shown by a simple calculation that
the shearing strength of the rivet is approximately equalled by the resist-
ance of the material to tearing, and thus the joint is not more likely to fail
in one direction than another. As plates increase in thickness with increase
in size of ships, the diameter of rivets become considerably reduced from
the maximum given above. Obviously, there is a limit to the size of rivet
which can be efficiency worked by hand, and when great strength is required
in a riveted joint and machine riveting is not available, this is obtained by
increasing the rivets in number rather than in diameter. A lower limit to the
size of rivets which may be used in any case is fixed by the punching machine,
which cannot punch holes of a diameter much less than the thickness of
the plates, as the punch is liable to crush up under the load. In the sub-
joined table we give Lloyd's Rules for the diameter of rivets in steel ships.
It will be seen that rivets 1^- inch diameter are required for plates 1 inch
in thickness or thereabouts. These heavy plates are usually restricted to
the keel or the sheerstrake, w T here ordinary machine riveting may be em-
ployed, but in the big Cunarders Lusitania and Mauretania^ the riveting of
a large part of the shell was done by special hydraulic machines with gaps
sufficient to take the full width of a strake, the strakes being fitted and
riveted up complete, one at a time, and consecutively.

Thickness of Plates in Inches.

■22 and
under "64

■34 and
under *48

•48 and
under "66

■66 and
under - 8S

■SS and
under 1 14

114 and
under 1*2

.Diameter of rivet in inches -

5
8

7
3"

Where joints are to be watertight, to ensure efficient caulking the rivet
spacing should not in general exceed 4 to 4J diameter, centre to centre. The
latter spacing, for example, is permitted in the joints of bulkhead plates,
in gunwale and margin plate angles, and elsewhere ; the former in the fore-

T38 SHIP CONSTRUCTION AND CALCULATIONS.

and-aft seams of shell-plating, the edge and end joints of deck-plating, in end
joints of shell-plating where these are quadruple riveted overlaps, in double
buttstraps, butts of margin plates, floor-plates and tie-plates, in the athwartships
and fore-and-aft joints of inner bottom plating, and elsewhere. In some posi-
tions, the need of providing sufficient strength entails the adoption of a closer
spacing than required for watertightness. Thus the rivets in the end joints
of shell-plating, where buttstraps are fitted, have a spacing of 3 J- diameters
in the rows parallel to the joints, and of 3 diameters in the rows at right
angles to the joints ; with overlapped end joints, except where they are quad-
ruple riveted overlaps, the rivets are 3 J diameters spacing in both directions.
Although the plate is thus pretty severely riddled with holes, there is still a
sufficiency of strength left in the material between them to prevent the
failure of the connection in that direction.

In a few places, where special reasons demand it, the spacing for water-tight
work is extended to five diameters centre to centre. The rivets in bar keels,
those in the angles of keels of flat plate type, also the rivets connecting water-
tight bulkhead frames to the shell plating, are of this spacing. In the case of
keels, as the rivets are large and the riveting is usually done by hydraulic
machines, which effectually close the surfaces, the spacing is found close
enough to obtain good caulking. In the case of bulkhead frames, although to
obtain watertightness a closer rivet pitch might be desirable, to resort to it
would accentuate the line of weakness through the frame rivet holes. The
standard weakest section is that through the ordinary frame rivet holes ; the
section in way of the closer pitched bulkhead frame rivet holes is made up to
this by doubling the shell in way of the outside strakes in the vicinity of the
bulkhead, or other compensation is made. When unhampered by the necessity
for caulking, we find that the spacing is widened. In the frames, beams,
keelson angles, bulkhead stiffeners, etc., the distance between rivets may be
seven diameters. In the case of frames, this spacing becomes modified in
certain circumstances. When the frames are widely spaced, viz., 26 inches
and upwards, to make up the number of rivets their spacing must be
reduced to 6 diameters. A similar reduction is required when the framing
is deep, viz., n inches and above, to develop its full efficiency. When
each frame consists of a channel bar and inner reversed bar, the spacing
must be reduced to 5 diameters, because the rivets connecting the frame to
the shell plating are liable to a relatively high stress, due to the neutral
axis of the frame girder being drawn towards the inner edge of the latter
by the reversed bar. •Experiments carried out by Lloyd's Register have borne
this out. In way of deep ballast tanks, and peak tanks, the spacing of
frame rivets is to be 5 diameters, and in way of the oil compartments of
bulk oil vessels, 6 diameters apart, owing to the circumstance that the
weight acts directly on the shell plating, and the strength of the framing is
brought into play entirely through the rivet connections.

RIVET HEADS AND POINTS.— There is much variety in the methods
of forming the heads and points of rivets ; a few of the commoner styles

RIVET HEADS AND POINTS.

139

are shown in fig. 140. The rivet marked A is known as the pan type, and
rivets employed in the shell, decks, tank top, and in handwork generally
where strength is the first consideration, are usually made thus: — The pan-
head is an efficient form, the shoulder under the head giving it good binding
power when riveted up. The rivet-head marked B is in favour with some
shipbuilders. It is considered that the plug-shape when hammered fits tightly
into the hole, and secures watertightness independently of the laying up
process; with pan-heads, such hammering frequently brings a strain on the
head without affecting the shank of the rivet. The plug-head rivet is some-
times used for decks, tank tops, etc., but owing to its lack of strength it
has not found favour for shell work. Clearly it has not the binding power
of the pan-head, and having little or no shoulder, under severe stress it is
liable to be drawn through the rivet hole. At G a snap form of head is

j^,

indicated. It is occasionally employed in handwork at places in sight, where
a nice appearance is desired, such as in casings, bulkheads, etc. The flush
head (see F) is only used in special cases where a surface clear of projections
is required. It is a somewhat expensive form to work, as, of course, the
plating must be countersunk to receive it, but it has fair holding power and
makes efficient work.

Of rivet points the commonest, and most efficient, is the flush one (see D).
In general, it is associated with a pan-head, as, for example, in shell work.
It is usually finished slightly convex, as shown, in order to maintain the
strength. Being flush, the holding power of the rivet has to be obtained by
giving the point the shape of an inverted cone. The widening of the hole
in the plate for this purpose is called countersinking, which entails the drill-
ing of each hole after punching. The flush point is sometimes adopted
where plug, snap, and flush heads are employed. Usually the snap head
goes with a snap point (see G). It cannot be said that this point, while it

14° SHIP CONSTRUCTION AND CALCULATIONS.

looks well, is always reliable. The manner of using the snap tool, with
which the point is finished off, is the cause of much of this ; frequently
the snap is applied before the rivet is thoroughly beaten into the hole, with
the result that many such rivets afterwards work loose. The snap style of
head and point is used in machine riveting, but the results are then invari-
ably satisfactory. This is, of course, due to the great pressure available, which
squeezes the rivet thoroughly into the hole and at the same time closes the
joint. The hammered point indicated at A and B is very efficient, as in
order to obtain the conical shape, it is necessary to subject the material
to a severe beating-up process, which causes the rivet to thoroughly fill the
hole, thus obviating the chief defect of the snap point.

At £ a tap rivet is illustrated. This type is really a bolt, and is used
where the point is inaccessible for holding up. It is frequently employed in
connecting shell plates to stern frames at the boss and at the keel.

In fitting tap rivets, the holes are first threaded, the rivet being then
inserted and screwed up with a key fitted to the square head on the rivet.
When the rivet is sufficiently tight the head is chipped off and the rivet
caulked.

In shipwork generally, the holes for rivets are made in the plates and
bars by punching. The positions are spaced off or marked from templates,
and are punched from the faying surfaces, i.e., the surfaces which come
together when fitted in place. One reason for taking this precaution is to
ensure that the rag, which is frequently left round the hole on the underside of
the plate or bar after punching, shall be clear of the jointed surfaces ; another
is to take advantage of the shape of the punched hole to increase the efficiency
of the riveting. As is well known, a punch in penetrating a plate makes a
cone-shaped hole, which has its smallest diameter at the point at which the
punch enters, and plates which are to be joined together are so fitted
that corresponding holes form two inverted cone frustrums, the finished rivet
having thus much greater holding power than if it were merely cylindrical.
Rivets are usually manufactured with cone-shaped necks to readily fill up the
space under the head (see fig. 140).

A disadvantage of punching holes in steel plates is found in the deteriora-
tion of the material in the vicinity of the hole which is thereby caused.
This deterioration takes the form of brittleness, the steel having thus a lia-
bility to break away when through stress the rivet bears upon it. When
rivet holes are countersunk this unsatisfactory material is largely removed. The
strength may also be restored by annealing the plates after punching, t\e. t
heating them to a cherry red and then allowing them to cool slowly.

Drilled holes are not largely employed in shipwork because of the greater
cost. Drilling, unlike punching, does not impair the quality of the material,
but the cone shape which is got by punching could only be obtained in
drilled holes by specially countersinking them. Sometimes, when considerations
of cost are not allowed to intervene, as at the sheerstrakes and upper deck
stringers of some large vessels, the holes are drilled in place by portable

STRENGTH OF RIVETED CONNECTIONS. I4 1

electric tools. By this means, perfectly concentric holes are obtained, and a
good quality of riveting thus assured.

STRENGTH OF RIVETED CONNECTIONS.— Riveted connections have
been frequently experimented upon with a view to obtaining the conditions of
maximum efficiency. Iron rivets are found to have maximum shearing strength
when in iron plates ; in steel plates their shearing strength is less — a f-inch
rivet, for instance, in iron plates has a shearing strength of 13*6 tons, which falls
to n£ tons in steel plates. This appears to be due to the increased shearing
effect of the harder plates upon them. Iron rivets are, however, much used
in steel ships, as they are more easily worked than steel rivets, and their
deficiency in strength is readily made up by increasing them in number. Steel
rivets in steel plates give excellent results when carefully worked. Indeed, the
quality of workmanship in riveted connections is of first importance.

A point worthy of note is the friction which exists between parts riveted
together. This is due to the contraction which takes place in the rivets
while cooling, causing the surfaces in contact to press on one another.
Careful experiments* have shown that the frictional resistance caused by
i-inch rivets, when the points and heads are countersunk, is 9*04 tons per
rivet, and by f-inch rivets, 4*95 tons ; with snap heads and points, the
results were — 1 -inch rivets, 6 "4 tons, f-inch, 472 tons. In hydraulic work
the frictional strength is greater. It is probable that connections are seldom
stressed beyond what can be resisted by the friction between the surfaces ; in
this case the rivets will not be under stress at all, and there will therefore
be no movement in the joints to disturb the caulking and cause leakage.

Of course, frictional resistance has its highest efficiency only when care
is taken in fitting the plates and in riveting them. When this is not done
the efficiency of a joint may be low indeed.

One fruitful cause of unsatisfactory riveting are blind or partially blind
holes. As mentioned above, rivet holes are usually marked from templates,
and the plates and bars are punched before being erected into place. Ob-
viously, to obtain an exact correspondence of holes with so many separate
processes is a most difficult matter, so that even in work of fair quality a
moderate number of holes are found out of line ; in careless work, the
percentage may be very large. When only slightly unfair, the holes may be
corrected by using a steel drift punch. This tool should not, however, be
driven into holes which overlap to any great extent, as the tearing of the
steel by the punch has a very pernicious effect upon it, much the same,
indeed, as that caused when punching the plates in the first instance, i.e.,
the material becomes brittle and liable to fracture under stress. The best
way to cure partially blind holes is to rimer them out to a larger size and
use rivets of increased diameter.

DECKS. — Next to the shell-plating, the decks are perhaps the most im-

* See an interesting paper by Mr. Wildish, in the Transactions of the Institution of Naval
Architects for 1885.

142 SHIP CONSTRUCTION AND CALCULATIONS.

portant features of a ship's structure. The top deck serves the purpose of
making the holds watertight and suitable for the carriage of perishable cargoes.
It is also available as a flat to walk upon, from which the crew may perform
the various operations required in working the ship. But, besides this, it
occupies a commanding position as a feature in the design. We saw, when
dealing with strains, that the top and bottom members of a beam are of great
value in resisting longitudinal bending tendencies, as they occupy positions
most remote from the neutral axis. In a beam or girder, such as the hull
of a ship, the top member is formed by the upper deck and the topmost
parts of the shell-plating ; the importance of this deck as an item in the
strength is therefore obvious. 'Tween decks, although they may not be
disposed so suitably as the upper one to resist longitudinal bending, are
yet splendid stiffeners of the hull, tying the sides together and offering
powerful resistance to racking tendencies. These intermediate decks are a
necessity in large passenger vessels, the sleeping and other accommodation
being provided in the space thus cut off below the upper deck. In cargo
vessels they are also necessary for some kinds of freight, although for general
trading purposes large holds without obstructions, such as intermediate decks,
are now much favoured (see Chapter V.)

The minimum number of plated decks required by an ocean-going steam-
ship, structurally speaking, varies according to her size, i.e., taking Lloyd's
Rules, according to her longitudinal number.

Of course, in small steamers and sailing vessels, no steel deck may be
structurally necessary, the strength being sufficient without it. In such a case
the necessary watertightness of the holds may be secured by covering the
beams with a wood flat or deck, caulking the seams with oakum and
paying them with pitch.

A wood deck has some advantages over one of steel. It has, for instance,
a finer appearance, and is pleasanter to walk upon, for which reason it is
always fitted in passenger vessels, even when a steel deck is required struc-
turally. For vessels trading in hot climates wood weather decks are desir-
able, as the effect of the sun's rays on unsheathed steel or iron decks is
such as to make it almost impossible to move about on them. The decks
of ordinary tramp steamers, however, are seldom wood-sheathed on account
of the cost, and because a steel or iron deck is found to stand much more
knocking about than one of wood, for which reason in many small cargo
vessels the upper deck is fitted of steel or iron, although uncalled for by
considerations of strength.

DECK DETAILS. — The most important part of the deck is the stringer;
indeed, all tiers of beams must have stringer plates riveted to their upper sur-
faces, whether a complete deck be fitted or not. These plates form a margin
strake to the deck, by means of which, through the medium of an angle bar,
it is connected to the shell plating. At weather decks this bar is continuous;
at intermediate decks the stringers are slotted out to allow the frames to pass
through, and the attachment to the shell is obtained by means of short, inter-

DECK DETAILS.

143

costal lugs between the frames, a continuous angle bar, however, being fitted by
way of compensation, along the stringer just inside the frames.

In fig. 141 the usual methods of fitting stringer plates at an upper and at
an intermediate deck are illustrated. As will be seen, in conjunction with the
shell plating, it forms a powerful T-shaped girder eminently adapted to resist
tendencies to deformation of transverse form. The upper-deck stringer plate is
specially important as affording considerable resistance to longitudinal bending.
The end joints of this strake must be at least double riveted, even in small
vessels; in larger ones, treble and quadruple riveting is essential; while in the
largest vessels, treble-riveted double straps are required. Both the latter
methods of forming a stringer end joint arc shown in fig. 142.

Fig. 141.

SECTION

UPPER DECK

PLAN

S~7

_l_q1 ~_"b_ _0 _

„L°]J IS*. l'_ .

L^j

SECTION
DECK BELOW UPPER DECK

PLAN

O J_o

'- V

flip

*— j

1°

lo
lo
lo

\°

1 3

L-^

The first plates to be fitted on a tier of beams are those of the stringer,
as they bind the structure together and keep it in proper shape. When a
wood deck only is to be fitted, the beams are further held to their work
by having two fore-and-aft lines of tieplates fitted, one on each side of the
centre abreast of the hatch openings, or in some other convenient position.
In sailing vessels severe racking stresses are communicated to the deck
through the masts, and to counteract these, a system of diagonal tieplates is
fitted in conjunction with those running fore-and-aft (see fig. 143). Where the
two lines of tieplates cross, to ensure that only a single thickness shall pro-
ject above the* surface of the beams into the wood deck, one is joggled under

144

SHIP CONSTRUCTION AND CALCULATIONS.

Fig.

142.

p o|i o io o o o| O

O O O OOIO o o

1
1

o o o 1
o o o o !
o o o o i

O O '

O O O J
O i
O o o '
O O ■

e-o-e -©-•<

O o o |
O u O j
O O I
O l
O I

>|o o ! o I l o o o o ! o c

H---1 -!-*■ j ~- - J

I ol! o

.■-j wwj ftwwwwwv ■ --■ - ,«r:-y

o o

o
o
o

o o

O i o o o

o
o

-©-,

o
o o

^wm._

o o
o

o o
° o

-o -o

o
o

o o

II o

I o

SECTION AT A.B.

SECTION AT CD.

Fig. US.

DECK DETAILS.

*45

the other, or is cut and a double-riveted lap or butt joint made. Tieplates
are of the same thickness as the stringer plates of the deck on which
they are fitted.

If a steel or iron deck is to be fitted, the tieplates are, of course, dis-
pensed with, and the deck plating, which is usually considerably less in thick-
ness than the stringer plates, is arranged in fore-and-aft strakes of considerable
breadth so as to minimise the number of rivet seams. The end joints of
deck plates in ordinary merchant vessels are invariably overlapped, and should
be double riveted for half length amidships, single riveting being sufficient at
the ends. The seams are usually single-riveted overlaps. When decks are
unsheathed, the end overlaps should be arranged looking toward midships, as

Fig. 144.

ELEVATION

-4

° °_l1ii 2 J? ° — ? _^_ _?

......

o o jo O O o o o o

L ^ . " i ^i

'o/[

I o o o

i O o J

SjK

SECTION

^COAMING
THICKENED PLATING.

l'o

PLAN

t « O O Q 'f o | * o o~o o o il°Tl

^CASING STIFFENERS

o o o o o Ho'o o o o o|o|

II Tf

DECK BEAMS

this allows of better drainage. For the same reason the fore-and-aft strakes
should be fitted clinker fashion, and the seams so placed as to impede drainage
to the scuppers as little as possible. When decks are to be covered with
wood, the clinker arrangement makes the fitting of the deck-planking more
difficult; in such a case, the raised and sunken system of deck-plating allows
of better work. In many recent cargo vessels- the edges of the unsheathed
steel decks are joggled, which obviates the fitting of slips at the beams, but
it has been objected that the depressions thus caused in the surface of the
deck form lodgments for drainage water.

At all deck openings compensation has to be made for the cutting away
of the material, the extent of this compensation depending on the strength

146

SHIP CONSTRUCTION AND CALCULATIONS.

required in the deck, and whether the latter has been fitted for strength
purposes, and not merely as a flat to walk on. In some cases increase in
thickness of the strake of plating alongside the openings is found sufficient;
in others, where the openings are very large, doublings are fitted. The strake
of plating alongside the machinery openings on the upper deck of large
vessels is an important one. Frequently, it is strengthened sufficiently so as
to make, when combined with a strong vertical coaming plate, a rigid girder
well adapted to resist longitudinal strains (see fig. 144).

The corners of large deck openings are particular points of weakness on
account of the sudden discontinuity of the deck-plating, etc., and unless pre-
cautions are taken there is a probability of fracture occurring at these points
when the vessel is under severe stress.

The usual precaution taken is to fit corner doublings as shown in fig.
145, but as well as this, at the upper deck within the half length, and also
within the same range at shelter, awning and bridge decks, if there be such,
the stresses being greatest at such places, girders should be fitted under the
decks in line with the hatch coamings, to which they should be efficiently

joined, or abreast of them, if not in the same line, in the vicinity of the
corners of the openings, so as to bridge over the weak points. Such arrange-
ments have been found to effectively strengthen vessels which had shown signs
of straining at the hatch corners ; they are now required by Lloyd's Rules.

Gutterways are usually fitted round the margin of weather decks where
these are to be laid with wood. They are formed simply by running an
angle bar fore-and-aft at a fixed distance from the ship's side, and riveting
it to the stringer plate. Steel gutterways are frequently coated with cement
as a preventative against undue corrosion.

WOOD DECKS. — In laying a wood deck, whether it be on top of one
of steel or iron, or merely on a tier of beams, it is important that tfre
planking should be fitted close down on the metal work. In way of the
tieplates and stringers in a non-plated deck, and of the edge seams and end
laps where a deck is plated throughout, the underside of the planking should
be scored out so as to obtain a solid bearing and an even upper surface.
With a plated deck oh the raised and sunken system, the planking fitted over
the sunken strakes is thicker than that over the raised strakes by the thickness
of the plating. Sometimes the difference in thickness is made up by slips of

WOOD DECKS. x 47

wood, but this is most objectionable, as spaces are thus left between the
wood and steel decks which a slight defect in the caulking of the wood
deck causes to become receptacles for the lodgment of water; when this
happens pitting and general decay of the steel deck quickly follow. Before
laying planks the steel work should be coated with a suitable preservative
composition, such as Stockholm tar powdered with Portland cement, and each
plank should be separately coated with tar before being bedded down. This
prevents the likelihood of lodgment spaces for water existing between the
metal and wood to cause decay. The best wood for weather decks is
undoubtedly teak, as it is of an oily nature and is well suited to stand
changes of temperature, but it is somewhat expensive. Pitch pine is frequently
used for weather decks of cargo vessels, where these are sheathed. It is
less costly than teak, but more of it is required, as a pine deck must be
thicker than one of teak. Pitch pine does not wear so uniformly, but it
is of a hard grain and fairly durable. Yellow pine makes a handsome
deck, and is therefore much used in passenger steamers. It is very soft and

Fig, 146,

therefore requires frequent renewals. On this account and because of its
extra cost, it is seldom used in cargo vessels.

Care should be taken when laying wood decks to have the hard side
of planks uppermost; this reduces the likelihood of the deck wearing into
holes in places. Three intermediate planks should separate butts in the
same frame space. The plank butts should be of vertical type {see fig. 146)
and arranged to come between beams where a steel deck is fitted, and on
the beams where there is no steel deck. They should be fastened with bolts
at each beam, or between the beams, where there is a steel deck ; and to
ensure that the planking shall lie perfectly flat, when it exceeds 6 inches in
breadth it should have double fastenings. Between 6 inches and 8 inches
broad, a bolt and nut and a screw-bolt is considered sufficient ; when the
planks are over 8 inches broad, two bolts and nuts are required. Deck bolts
should be galvanised, and should have their heads well bedded in white lead,
with grommets of oakum. When screwed up from below, the heads should
be sufficiently sunk in the deck to allow of a dowel being fitted over the top.

Pine decks should not be laid until the wood is thoroughly seasoned.
If this precaution be not taken, the deck is likely to open at the seams and

148 SHIP CONSTRUCTION AND CALCULATIONS.

become leaky as well as unsightly. Lloyd's Rules require a period of four to
six months to elapse, according to thickness, from the time of cutting to the
time of using. Pitch pine planks for weather decks should be seasoned for
six months. The above periods of seasoning are not required where a satis-
factory artificial method of seasoning is adopted.

It should be mentioned that wood decks are now being superseded in
passenger and crew spaces by compositions like litosilo, corticene, etc. Decks
thus covered are comfortable to walk on, have a good appearance, and, when
carefully laid, have generally been found to wear well.

CARGO HATCHWAYS.— There must be at least one deck opening into
each main compartment of a vessel to allow of cargo being shipped into,
and discharged from it. In the latest cargo steamers these hatchways are
of considerable size, so as to be suitable for special cargoes of large measure-
ment, such as pieces of machinery. Lengths of 24 to 28 feet, and breadths
of 16 feet, are common, while these dimensions have frequently been exceeded.

We have already indicated some of the means adopted to prevent these
large gaps in the deck from becoming dangerous points of weakness, and it
now remains to show how the hatch openings are framed.

The main portion of this framing consists of vertical coaming plates
fitted fore-and-aft and athwartships and carried down to the lower edge of the
deck beams, those in a fore-and-aft direction forming an abutment for the
beams that have been cut, and those fitted athwartships stiffeners to the con-
tinuous hatch-end beams, to which they are securely riveted. The connec-
tion between the hatch coamings and the severed beams is effected by
means of angle lugs, fitted single where beams are at every frame, and double
where they have a two frame spacing (see fig. 147). Lloyd's Rules require
that there shall be three rivets in each flange of these lugs when attached
to beams 7 -J- to 9^- inches deep, the number being increased to four where
the depth of beam is 10 to 12 inches.

The deck-plating is fitted so as to abut against the coamings, a riveted
attachment being secured by means of a strong angle bar. In non-plated
decks broad tieplates are fitted on the beam ends and against the coamings,
and in this way a strong T-shaped girder is obtained round the edge of
the opening. It should be mentioned that when decks are laid with wood,
the vertical flange of the hatch-coaming bar is fitted of sufficient depth to
project h inch above the wood, so as to facilitate the caulking of the latter.

Weather deck hatch coamings (see fig. 147) should be of considerable
height above the deck so as to protect the comparatively weak covers which
seal the openings from receiving the full force of the heavy seas which in
rough weather frequently fall upon the deck. On upper decks, coamings should
have a minimum height of 2 feet except under awning or shelter decks. In
certain classes of vessels, which have deep wells between the front of the bridge
deck and the aft end of the forecastle on the upper deck, the coaming
height should not be less than 2 feet 6 inches, as these spaces are specially
liable to flooding. Obviously, on bridge, awning and shelter decks, which are

CARGO HATCHWAYS.

149

situated high above the water/line, hatch coamings may be of reduced height;
they need not, in fact, exceed 18 inches.

Weather deck coaming plates, in order to be efficient as girders and as
protecting walls to the hatchways against inroads from .the sea, should be of
a substantial character. For instance, the coamings of hatches, under 12 feet
in length should be -36 of an inch thick, while those having lengths of 16
to 24 feet should have side coamings '44 of an inch thick; end coamings

Fig. 147.

ELEVATION

' ' I

ftfr

"IT

-tf-T~ir

nr*

ir irijf

. ^ PUN

ALTERNATlVe PLAN
FOR HATCH BESTS

in the larger hatchways are allowed to be '04 of an inch less in thickness
than the side coamings, owing to their shorter length, and to the fact that
they have no beam ends to support like the side coamings.

Below the weather deck hatch coamings need not be so deep, as, of
course, the openings are not exposed to the sea, and high coamings would
impede the efficient stowage of cargo in the 'tween decks (see fig. 148). In
'tween deck hatches 10 feet and under 14 feet long, the total depth of side
coamings from underside of hatch end beams may be 16 inches, and in

'5°

SHIP CONSTRUCTION AND CALCULATIONS.

those from 18 feet to 24 feet, a depth of 20 inches is considered sufficient.
The loss of strength due to reducing the depth of the hatch coamings at
lower decks is made up to some extent by increasing the thicknesses by '04
of an inch, as compared with upper deck hatches of the same length.
Round corners are usually preferred for hatches on weather decks. This
style makes the fitting of the wood covers at the corners somewhat difficult;
it is also less convenient for fastening the tarpaulins, but otherwise it has
obvious advantages. To begin with, the tendency for the deck to strain at
the hatch corners is less where these are round than where square. Round

Fig. 148.

ELEVATION

DETAIL ATA..

' 3Ji«3t»<46

DETAIL AT CO.

7-T.y-r--'

SECTION AT A.B.

KJl-j'iTTi.r'ir SgSSSSSZZ: .^-"^^l^JJJv T-.--TT7^

j l fff— TT ft"

DETAIL AT HATCH CORNER

L VERTICAL FIANCE
CUT AWAY

corners are less likely to damage cargo which may collide with them ; they
have also a nice appearance. The same advantages of having round corners,
obviously, do not extend to 'tween deck hatches and, consequently, they are
usually of square type (fig. 148).

In order to strengthen hatch coamings against inroads from the sea
and to provide adequate support to the wood covers, portable athwartship
beams are fitted. In hatches 10 and under 16 feet long, one such beam,
formed of a plate with double angles at top and bottom, or other equivalent
section, is required; in those of 16 to 20 feet in length, the portable beam

CARGO HATCHWAYS.

151

becomes a web-plate extending to the bottom of the coamings, fitted with
double angles top and bottom. Two web-plate girders of this description
are required in hatches from 20 to 24 feet in length. These portable
beams are frequently bolted between double angles riveted to the coamings,
when they .act as ties as well as struts, and to some extent compensate
for the gaps in the deck made by the hatch openings ; occasionally, they
are arranged to ship into special shoes.

Fig. 149.

SECTION AT AB

■WATCH COAMING STFTE.NqTHENE.0
»N LlLU OP PILLARS

SECTIOM THRO C

MAI.' OlAMr

SECTION THffO*EP,

Web-plate beams in hatchways below the upper deck should be equal
in thickness to the coamings to which they are attached, and should extend
to the lower edge of the coamings. Where the latter are shallow, as in the
case of 'tween deck hatches, the web-plates are to be a quarter deeper in
the middle than at the ends, and stiffened top and bottom by double angles.

The top angles of the portable webs, as already hinted, form lodgments

1^2 SHIP CONSTRUCTION AND CALCULATIONS.

for the wood covers, but, as previously mentioned, heavy seas frequently fall
upon the deck, and the covers have to sustain a substantial share of the
weight; they therefore need additional support. This is provided by fitting
strong steel or wood fore-and-aft bearers. In small hatches from 6 to 10
feet broad, a single bearer fitted at the centre is sufficient; in larger hatches
three or more in the breadth are required. The fore-and-afters should fit
into iron or steel shoes securely riveted to the end coamings and to the inter-
mediate webs, if any, and the shoes should afford a bearing surface not less
than 2 inches broad. To support the wood covers at the hatch sides, ledge
or rest bars are fitted, giving a bearing surface at least if inches broad.
This ledge iron is riveted to the hatch coamings between the webs, except
where the side moulding and ledge rest consist of a single special section,
when it is, of course, continuous for the length of the hatch (see fig. 147).
As well as at the top of coamings, mouldings are frequently fitted at the
bottom on the inside to take the chafe of cargo. The latter requirement
is sometimes met by flanging the lower edge of the side coamings, instead
of fitting mouldings. At weather deck hatches, to ensure watertightness,
strong tarpaulins are fitted over the wood covers, usually two or three to a
hatch, one placed above another. The tarpaulins are secured in position
by means of flat iron bars wedged into cleats riveted to the hatch coamings,
round which they are spaced about 24 inches apart.

In many recently built cargo vessels hatch beams have been fitted all in
one direction, i.e., either all athwartships or all fore-and-aft, the direction being
that of the shorter dimension of the opening, which, in ordinary cases, is
athwartships. Lloyd's rules now provide for arrangements of wholly transverse
webs for hatches ranging in breadth from 12 to 20 feet. The supports at
the coamings for the wood covers in this case should have a bearing surface
3 inches broad. Fig. 149 shows a hatch framed in this way.

HATCHWAYS INTO DEEP TANKS.— These should be strongly framed
and have means of closing in a watertight manner as they must withstand
the testing pressure on the tank, viz., an 8-feet head of water above the
crown of the tank, without straining or showing a leak. To simplify con-
struction, watertight hatchways are made no larger than absolutely neces-
sary. Usually they are about 6 feet to 8 feet square and, as already
pointed out, owing to .the presence of the middle-line bulkhead with which
deep tanks are provided, are fitted two abreast. The coamings of these
hatchways frequently consist of deep bulb angles (fig. 150), but sometimes
they are built of plates and angles. The cover is a plate of substantial
thickness, with angle or bulb angle stiffeners at about -» feet spacing. It is
secured in position by nuts and fall back bolts, or nuts and through bolts,
and watertightness at the joint is effected by packing it with spun yarn or
rubber. To gain admission to the tank without removing the hatch cover,
a watertight manhole door is usually fitted in the latter.

CARGO PORTS AND DOORS.— Many vessels are fitted with small side
ports to give access to the 'tween decks. These are found useful in load-

Cargo PORts and books.

153

ing certain classes of bale goods, and allow the 'tween decks to be stowed
while the holds are being filled through the main hatchways. When side
ports do not exceed, say, 3 feet square, sufficient compensation for cutting
the opening in the shell-plating is provided, by doubling the strake above
it for a short distance and fitting a stout angle around the edge of the

Fig. 150.

SECTIONAL ELEVATION

DETAIL OF FASTENING

ALTERNATIVE METHODS
OF SECURING COVER

SPUN YARN

RUBBER

BOLTS SPACED
" 6" APART

opening. The door is sometimes secured by bolts and nuts at sufficiently
close spacing to ensure a watertight joint with canvas and red lead between
the surfaces ; usually, however, strong backs are fitted inside, one or two
being used according to the size of the door; and to obtain watertightness
at the joint spun yarn or rubber packing is used.

154

SHIP CONSTRUCTION AND CALCULATIONS.

In certain vessels — in those engaged in the cattle trade, for instance —
very large doors are fitted in the ship's side in way of the bridge or
shelter 'tween decks. These doors make big gaps in the side-plating and
have to be carefully compensated for. Usually the shell-plating is doubled
above and below the opening and for some distance, say two frame spaces,
beyond each end of it; web frames are also fitted in the 'tween decks at
each end of the doorway. The shell doublings make good the longitudinal

Fig. 151.

ELEVATION

SECTION THRO
DOOR

DOOR
FRAME

DOOR NAME
6*6x-60"

strength, and the web frames restore the loss entailed in the cutting of the
side frames in way of the opening. Fig. 151 gives details of a cattle door
12 feet long and 5 feet 6 inches deep, as fitted in a large modern cargo
and passenger steamer.

DERRICKS AND DERRICK POSTS.— Large and numerous hatchways
are of little value unless an efficient installation of appliances for working the
cargo in and out of them be also provided. This is specially the case with

DERRICKS.

*SS

steamers whose economical working demands the utmost despatch in the load-
ing and unloading of cargo. Sailing ships usually make long voyages and are
seldom in port; they can, therefore, afford to spend a longer time there than
the more ubiquitous steamer. Moreover, their working expenses are much
less than those of the latter. For these reasons an expensive system of cargo
gear is seldom fitted in sailing vessels. Hand-power winches are considered
sufficient, and the cargo gear is usually suspended from the lower yards or
from convenient wire spans.

The cargo gear of modern steamers may consist of (i) ordinary derricks
with steam winches, (2) hydraulics derricks, (3) steam cranes, (4) electric cranes.
Electrical appliances, although frequently proposed, have not yet come much
into use. Steam cranes are frequently fitted in coasting vessels, as they hoist

Fig. 152.

ELEVATION

and slew quickly, and thus minimise the time a vessel need remain in port
— an important consideration where a vessel has to be loaded and discharged
every day or two, or even more frequently. Moreover, steam cranes may be
placed anywhere about the deck. They take up a great deal of room, how-
ever, and are more expensive than steam winches and derricks, for which reasons
they are seldom fitted in ordinary ocean-going cargo vessels. Hydraulic der-
ricks are sometimes fitted on first-class passenger steamers, as they work
smoothly and without noise. They are costly to install, as, of course, a power-
ful pumping engine is required in order to maintain an artificial head of water.
The system of working cargo almost universally adopted in ordinary cargo
vessels is that comprising steam winches and ordinary derricks. The latter
may be constructed of wood or steel ; if for small lifts, say, from three to
five tons, pitch-pine derricks are commonly fitted. They are hinged, if practic-

156

SHIP CONSTRUCTION AND CALCULATIONS.

able, on the masts, which, in cargo steamers, are now little else than derrick
posts. One derrick and winch per hatch is sufficient where the holds are of
moderate size ; where they are large, however, and where loading or discharging
can be carried out on both sides of the vessel at once, two derricks and

Fig. 153.
OUTREACHlS FOR DERRICK. SPANS

PLATING "50

EYEPLATES FOR SPANS
l\ MOULDING

DERRICK

TABLES

winches are necessary. We thus see that in the usual arrangement, with a
mast between two hatches, the former may have to support four large derricks
with their respective loads. In special cases, where separate derricks are
employed for hoisting and for slewing, the number of derricks per mast may
exceed four. In fig. 152 is shown the usual way of hinging derricks on masts.

DERRICKS AND DERRICK POSTS. 1 57

Steam winches must be placed with careful regard to the derricks.
Single winches are, of course, situated at the middle line with the middle of
the winding drums in line with the derricks. Double winches should be
placed on each side of the centre of the ship, with a sufficient distance be-
tween them to allow a man to pass. To obtain compactness, the inner drums
are frequently dispensed with, and to ensure direct leads from the derricks
to the winding barrels, the axis of the winches are inclined to the middle
line. Frequently, it is preferred to have the winches square to the middle
line, as the winchmen are then better able to observe operations ; in these
cases, direct leads to the barrels are obtained by means of snatch blocks on
the decks, or, better, by extending the derricks out transversely on tables
(fig. 153), so as to come in line with the middle of the winches. With
this arrangement it is desirable to have the point of suspension in each case
immediately over the heel of the derrick, otherwise difficulty will be experienced
in slewing the latter. This drawback is found, for instance, where a single
derrick is worked from a mast having considerable rake, and no arrangement
is made to bring the point of suspension over the heel of the derrick. If the
derrick is fitted forward there is a strong tendency for it to lie overboard,
and if aft on the mast, to lie over the middle of the hatch ; considerable
power being required to slew the derrick, particularly if loaded, against either
of these biassed directions. The advantage of plumb derricks is therefore
obvious, and some vessels are built with vertical masts to this end.

When a mast is situated too near a hatch to allow of a derrick hinged
on it being sufficiently long to plumb the centre, and swing clear of the
ship's side, it becomes necessary to resort to the use of derrick posts. These
may be placed between the hatch and the ship's side, and with a com-
paratively short derrick a sufficient outreach may be easily obtained. Derrick
posts were at first objected to as being unsightly, but their great utility has
outweighed considerations of this sort, and they are now to be found in
many up-to-date cargo steamers. Where the lifts are at all great,, derrick posts
should be made of considerable height, otherwise excessive stresses will be
brought upon them, as well as on the derricks and the spans. This can be
readily demonstrated by drawing a diagram of forces. Derrick posts to carry
a 10-inch to 12-inch derrick, and lift ordinary cargo, should be 20 inches to
24 incises diameter at the deck, and have a height of 24 to 28 feet. Large
derrick posts are constructed of |-inch steel plates, a little taper being allowed
in the thickness towards the top. To give rigidity, a housing equal to the
height of one 'tween decks is desirable; where this cannot be obtained, deep
brackets must be fitted to the deck. Fig. 154 shows a derrick-post and
appurtenances as ordinarily fitted.

SEATS FOR STEAM WINCHES, Etc.— As before mentioned, decks should
be stiffened locally in way of steam winches, by fitting plating on the beams,
and by supporting the latter by special pillars. If winches are to be placed
on a wood deck, the part under each winch should be of hardwood, or the
wood deck increased in thickness locally, as the wear and tear is great at

i58

SHIP CONSTRUCTION AND CALCULATIONS.

these places. Sometimes a steel angle bar is fitted round the winches and

riveted to the deck-plating, the wood deck being butted against this bar, and

the spaces enclosed coated with cement or left bare. In such a case, the

Fig. 154.

winch sole plates are either bolted directly to the steel or iron deck, or
raised on Z or channel bar stools. The latter plan is the better one, as
the stools stiffen the deck and take most of the vibration caused by the

SEATS FOR STEAM WINCHES.

J 59

working of the winches. In ordinary cargo steamers, which have, as a rule,
unsheathed decks, winch stools like the above are commonly fitted.

The steam supply and exhaust pipes to the winches (where the exhaust
steam is returned to a tank in the machinery space) are usually led from the
machinery-casing along the deck just outside the line of the hatchways, and
are supported on stools of cast or wrought iron (fig. 155), except where
they can be conveniently held by clips riveted to the casings or to the hatch
coamings. Sometimes separate exhaust pipes are led from each winch across
the deck to the ship's side. This latter plan has only cheapness to com-
mend it, as the cloud of escaping steam always present about the deck
during loading or discharging operations, is most objectionable.

To protect the winch pipes from damage, solid plate or sparred iron
covers are fitted over them.

MASTS. — In a sailing-ship the masts are probably as important as
the hull itself, since her power of locomotion depends on them ; in a
steamer they have a much reduced value, and are even not indispensable,

Fig. 155.

some classes of steamers having none at all. As seems fitting, therefore,
"we shall first consider the masts of a sailing-ship, and afterwards indicate
the modifications usual in the case of a steamer.

In a modern sailing-ship of average size, the masts, like the hull, are
constructed mainly of steel, their diameters and scantlings being graduated
in accordance with the strains they may be called upon to bear through the
action of the wind-pressures on the sails. As the mast bending moments
vary with the lengths of the masts, length is the natural basis on which tb
fix scantlings, and, in compiling tables of the latter, this method is usually
followed. Taking Lloyd's Rules, a lower mast 60 feet long has a maximum
diameter of 20 inches, with plating -fa inch thick, and one 96 feet long
a diameter of 32 inches, and a plating thickness of -|£ inch, the scantlings
of masts of intermediate lengths lying between these.

Mizen-masts of barques carry no cross-yards, and support a less sail area
than mainmasts; reduced diameters and scantlings are therefore allowed in
their case.

The maximum diameter of a mast and the greatest thickness of plating
are at the deck, as the bending moment is obviously greatest there ; towards
either end, the diameter and the thickness of the plating are somewhat

i6o

SHIP CONSTRUCTION AND CALCULATIONS.

reduced. The number of plates in the circumference of a mast is governed
by practical considerations. Lower masts, with a rule length of 72 feet and
under, are built with two plates in the round ; those above this length should
not have less than three plates. These plates are usually overlapped at edge
and end joints ; sometimes the latter are butted, in which case the straps
should be fitted outside, as opening at the joints due to bending of the masts
is thus prevented, and better work can be made in the fitting of internal angles
where these are necessary.

Fig. 156.

ELEVATION.

PLAN.

As the principal stresses on masts are cross-breaking ones, the end joints
are very important, and in all cases should be treble-riveted above the
deck; in way of the housing — by which is meant the part of the masts
below the deck — double-riveting is permitted. The seams should be double-
riveted ; but in masts under 84 feet long, single-edge riveting is considered
sufficient, provided the loss of stiffening effect due to reducing the lap is
made good by fitting internal angle bars. Above 84 feet length, double-
riveted seams are required as well as internal stiffening angles, as the bending
moments on masts of this length may be very great.

Rigidity is given to masts by securely fixing them into the hull, and stay-
ing them by means of steel wire ropes. Fig. 156 shows the modern method of

MASTS.

161

framing a mast-hole and wedging the mast at the upper deck — the deck at
which this is usually done. As will be observed, a stout plate is fitted on
the beams, which, in non-plated decks, must have a breadth equal to three
diameters of the mast. This plate is riveted to the beams and (in non-plated
decks) to diagonal tieplates (fig. 143), which unite it to the side stringers and
distribute the stresses communicated from the mast through the wedging.
A bulb angle ring about 4 inches greater in diameter than the mast is
riveted to the deck-plate, and when the mast is shipped the space between
this ring and the mast, the plating of which should be doubled in this
neighbourhood, is tightly wedged with hard wood. Above the deck the
wedging is neatly rounded, and a canvas cover or coat, usually double, is bound
to the mast and over the ring to prevent leakage of water into the hold space.
The doubling of the mast at the deck is to give strength, but particularly
to compensate for corrosion and pitting which may take place in way of the
wedging, the material being there inaccessible except when the vessel is
under special survey. When the mast-plating is doubled, the wedges need not
be removed before the third special survey, i.e. about every 1 2 years.

At the heel the mast should be supported on a strong stool, as very
great downward stresses are communicated to the mast through the rigging,
and if due provision be not made to resist these the mast may be forced
downwards, the plating at the heel crushing up or the stool collapsing if
not efficient Such movement of the mast would cause the rigging to be-
come slack and valueless as a support against bending.

Where a mast is stepped on a centre keelson, a good stool may be
contrived by fitting a strong plate immediately under the mast, and support-
ing it by brackets connected to the keelson and floorplate on each side of
the middle line. For wedging purposes a ring is fitted on the plate round
the mast-heel, and to keep the mast from turning, an angle or tee lug, riveted
to the plate, is fitted through the bottom of the mast. The mast-heel plating
is usually doubled for about 2 feet up from the bottom.

The main portion of a mast, and that upon which the principal dia-
meters and scantlings are fixed, is known as a lower mast, but above this
there are a topmast, a topgallant mast, and a royal mast. These upper spars
are sometimes constructed of wood, but in modern sailing vessels of fair
size steel topmasts are common. Lower masts and topmasts are occasionally
built as single tubes, but usually they are separate, the union between them
being effected by overlapping in the manner indicated in fig. 157. In this
case the topmast, which is of wood, passes through a cap hoop at the lower
masthead, and is supported by a rectangular bar of iron, or fid, which passes
through the heel of the topmast and rests on strong cheek-plates riveted
to the lower mast. This overlapping method is sometimes adopted for
uniting the topmast with the upper portions, the topgallant and royal masts
usually consisting of a single wood spar ; but where the topmast is of steel
the upper spar is frequently housed into its upper end.

The scantlings of steel topmasts, like those of lower masts, vary with

l62

SHIP CONSTRUCTION AND CALCULATIONS.

the length. Topmasts 38 feet long and above should have internal stiffening
bars. The edge seams of topmast plating may be single-riveted, but the end
joints, like those of lower masts, and for the same reason, should be treble-
riveted.

Obviously, so tall and comparatively slim a structure as a mast such
as we have described, must be strongly stayed in order to hold it to its work
of supporting the cross-yards and sails and resisting the wind pressure. We
have mentioned that steel wire ropes are used for this purpose. Lower
masts are stayed laterally by shrouds, which loop round the mast at the

Fig. 157.

PLAN OF TOP.

hounds and extend down to the gunwale, where they are attached to chain
plates riveted to the sheerstrake. Shrouds of smaller size are also fitted to
the topmast and top-gallant mast. These are not carried down to the ship's
sides, but are fastened to the mast just below the lower mast-top and the
topmast trestle-trees respectively, the necessary spread being obtained by
means of the mast-top and topmast cross-trees. As well as shrouds, the
upper spaces are further held by backstays fastened to the gunwale and to
the mast. In a fore-and-aft direction the masts are stayed to one another
by powerful wire ropes at various heights. The stays of the foremast are
run down to the forecastle deck, the upper ones being attached at their
lower ends to a bowsprit ; this arrangement, which is to allow of sufficient
spread in the stays, also permits of large-sized staysails.

BOWSPRIT. 163

Obviously, all this rigging will have little staying value if it be slack,
as in that case the mast which first receives the stresses due to the wind
pressure might break before the strength of the wire could be called upon.
Cases are on record of masts collapsing through lack of efficiency in the
stays in this respect. To obviate such disasters, the standing side rigging of
all vessels should be provided with rigging screws of simple design, by means
of which the shrouds and backstays may be readily tightened up at any time.

BOWSPRIT. — In modern sailing-vessels of fair size this spar is con-
structed of steel, and as it has to withstand considerable bending stresses,
due to the pull of the maststays attached to it, it is built of substantial
diameter and thickness of plating; the latter, indeed, is about the same as
for a lower mast of equal diameter.

Usually the bowsprit is housed in the forecastle, passing through an
aperture in a transverse bulkhead, or knighthead plate, fitted at the fore-end
of the forecastle, and abutting against a vertical plate extending between the
upper and forecastle decks, and strongly bracketed to the main deck-plating.
To secure it in position, the bowsprit is wedged in way of the knighthead
plate, angle rings being fitted to the latter around the aperture to take the
wedging. Internal stiffening angles are fitted in the middle of each plate
in the round, and, in addition, when the spar exceeds 28 inches in diameter,
a vertical diaphragm plate is fitted in way of the wedging and extended
some distance either way beyond. The end joints of the bowsprit plating
outside the wedging should be treble-riveted ; inside the forecastle, they may
be double-riveted.

Sometimes, instead of being housed in the forecastle, the bowsprit is
sloped away on the lower side at its after-end and bedded on the forecastle
deck-plating, to which it is securely connected by strong angle bars ; the
forecastle deck being stiffened in this neighbourhood by fitting the beams
on every frame. The outer part of a bowsprit when fitted as a separate
spar is called a- jibboom. The latter is usually built of wood and is fitted
through the- cap-band of the bowsprit. Frequently, in large modern sailing-
ships the bowsprit and jibboom are made of steel in one length, when it
is known as a " spiked bowsprit."

The bowsprit is stayed laterally by means of wire shrouds, and strong
bobstay bars are fitted to eye attachments on the stem and the underside
of the cap-bands at the fore-end of the bowsprit and jibboom.

YARDS. — The cross-yards of small sailing-vessels are constructed of wood,
usually pitch pine. In large ships having steel masts, the lower yard and the
one above it are frequently built of the same material as the mast. The
greatest diameter of a yard is, of course, in the middle, and is taken at -^
of its length ; at the ends it is tapered to half of this. When built of steel,
yards have single-riveted seams and treble-riveted end joints. The lower top-
sail and lower topgallant yards on each mast are usually fixed to the latter,
but with attachments designed to allow of free movement to any angle ; the
upper topsail and upper topgallant yards are attached to parrel hoops which

164 SHIP CONSTRUCTION AND CALCULATIONS.

fit loosely on the mast, thus admitting of each yard being hoisted into
position by means of appropriate running gear.

A special feature in the masts and yards of sailing ships are the mount-
ings. These are very elaborate and are made of great strength, as the safety
of a vessel might be seriously threatened if even one stay attachment were
to give way, on account of the increased stress which would thus be brought
on the others.

MASTS OF STEAMSHIPS.— The masts of a steamer do not call for
much comment. As has been said, in most modern cargo steamers they are
mainly fitted for ornament; incidentally, they can be usefully employed as
derrick-posts and standards for signal lamps, etc. The main function of masts
in sailing-ships, which is to carry sails, has been almost done away with in
steamers. No yards or square-sails are now carried ; two small fore-and-aft
sails on each mast are all that are usually arranged for, and these are fitted
not for propulsion but to give steadiness in rough weather. In many
modern cargo vessels even these are omitted. This omission of sails has
been deplored in some quarters, and there certainly seems a lack of economy
in neglecting to use the power of the wind for propulsion when it is avail-
able. The steadying effect of sails when a vessel is in ^a seaway is well
known.

Owing to their auxiliary character, a steamer's masts are of smaller
diameters and scantlings than those of same -length in a sailing-ship. Where
fore-and-aft sails only are carried, Lloyd's Rules permit the diameters to be
a fifth less, and the plating of a thickness to correspond with this reduction.

It has been pointed* out that in the case of a steamer's masts, whose
main duty is to withstand the strains due to the working of derricks, the
breadth of the ship should be considered in fixing upon the diameters and
scantlings. The broader a vessel, the greater will be the outreach of the
derricks, and, therefore, the greater the bending stresses on the masts. At
present no notice is taken of this in Rules for masts, and in the case of
a very broad vessel, where the mast has, say, to support four derricks, they
are frequently none too strong for their work.

It is customary to make a steamer's masts of pole type, i.e., in one
piece from heel to topmast head. As the sail spread is unimportant, no
greater height than this is necessary, so that topgallant and royal masts are
dispensed with, the finishing pole being fitted into the topmast. Frequently,
the topmasts are of wood, and made to ship telescope fashion into the
upper ends of the lower masts, appropriate gear being provided for the pur-
pose. Such an arrangement is demanded to allow of the vessel passing
under bridges in reaching ports like Manchester.

The edge seams of the mast-plating may be single-riveted, but the end
joints, like those of a sailing-ship's masts, must be treble-riveted above the

* See a paper by Mr. W. Veysey Lang, read before the Institute of Marine Engineers
in February, 1909.

MASTS OF STEAMSHIPS.

165

deck or partners, and double-riveted in the housing. When the masts are
of considerable length, the strength should be augmented by fitting and
securely riveting internal angle bars up the middle of each plate in the
round. The masts must be supported athwartships and fore-and-aft by strong
steel wire standing rigging.

The usual arrangement is, say, three or four shrouds, immediately abreast
the masts on each side, with two fore-and-aft stays. This is not the best
arrangement for the purpose intended. Instead of the close-spaced shrouds,
it is better to fit two with as great a spacing as possible, as much, in fact,
as will still permit the derricks to swing clear of the side. For access to
the masthead an iron ladder may be riveted to the mast, or two additional
shrouds may be fitted at close-spacing with ratlines to the masthead.

The need of strong work at the mast-heels has been pointed put in
the case of sailing ships, and similar remarks apply to steamers; for as well

Fig. 158.

as the bending stresses already referred to, the working of the derricks
give rise to considerable downward thrusts, steamers should therefore be
strengthened in way of mast steps. If these come on an inner bottom,
brackets should be fitted to the centre girder, unless the mast-heel happens
to be immediately over the junction of a floor-plate with the centre girder.
Fig. 158 shows the arrangement when a mast is stepped on a tunnel. The
stiffening of the latter in way of the step, which usually consists of stout
angle-bars riveted to the plating, is not shown in the sketch. A steamer's
masts are usually wedged at the upper-deck, and the arrangement is very
similar to that described for a sailing-vessel.

BULKHEADS. — This name is given to all vertical partitions, whether
fore-and-aft or athwartships, which, in a ship, separate compartments from
one another. Many "of these partitions are of little value structurally, as
those of wood between cabins, or those which, though built of iron or steel,
are only intended to act as screens and are therefore of the lightest de-
scription. Bulkheads, however, which divide a steel vessel into watertight com-

1 66 SHIP CONSTRUCTION AND CALCULATIONS.

partments, are of immense importance structurally and otherwise, and it will,
therefore, be of interest to consider their principal functions and the lead-
ing features in their construction.

These main partitions, which are usually placed transversely, are strongly
built and made watertight, so that in the event of a compartment being
filled with water the containing bulkheads shall be strong enough to support
the pressure and have joints tight enough to prevent the fluid escaping
into adjoining compartments. Generally, main watertight transverse bulkheads
are of value for the following reasons : —

i. As Elements of Strength. — Where they occur the hull is prac-
tically rigid transversely, so that they effectually prevent any tendency to
deformation in that direction; they also afford support to the longitudinal
framing, />., to the side stringers and keelsons, when the latter are
efficiently bracketed to them. The keelsons, indeed, of themselves have but
little rigidity, but when braced to the bulkheads they become efficient as
girders, and transmit the stresses brought upon the hull to the massive
bulkheads, thus spreading the strength of the latter over the region of the
structure lying between them.

2. As Safeguards Against Spread of Fire. — Their importance in this
respect can hardly be over-esUmated, isolating as they do the various holds
with their contents from each other. Many instances are on record of
vessels having been saved from total destruction by fire through the medium
of their bulkheads.

3. As Preventatives Against Foundering Consequent on the Pierc-
ing of the Hull by Striking a Rock or Otherwise.— We are already
familiar with the effect on the flotation of bilging a compartment, and have
seen that if the latter be large the loss of buoyancy may be sufficient to sink
the vessel. The importance of restricting the lengths of compartments by
a sufficient number of watertight bulkheads is therefore obvious.

In the case of a steam-vessel, there are certain conditions which fix the
lower limit of the number of bulkheads required. With the machinery
amidships, for instance, there should be at least four : one at a short dis-
tance abaft the stem, one at each end of the machinery compartment, and
one placed at -a reasonable distance from the sternpost. With the machinery
aft, a minimum of three watertight bulkheads might be allowed, the after-
bulkhead forming one end of the machinery compartment.

Of the above divisions the forward one, which is fitted as a safeguard
in the event of collision, is probably of chief importance. It should not be
placed too far aft, or the loss of buoyancy due to bilging the peak com-
partment may be sufficient to cause the vessel to go down by the head.
Lloyd's Rules require it to be fitted at a twentieth of the length from the stem,
measuring at the height of the lower deck. The collision bulkhead, as it is
called, has proved of immense service in saving vessels, and has often en-
abled them, though seriously damaged by collision, to make a port in safety.

It may here be said that the collision bulkhead is the only one usually

BULKHEADS. 167

fitted in sailing ships. In this case the transverse strength is made up
otherwise, and the question of cost, to mention no other, has put to one
side any idea of fitting numerous bulkheads.

The importance of the bulkheads which isolate the engines and boilers
from the cargo spaces scarcely needs emphasis. The chance of fire and
other damage to cargo, if there were no efficient tight divisions, is clearly
apparent. It is also necessary that the machinery compartment should be
quite self-contained, so that the bilging of neighbouring compartments would
not mean the extinguishing of the boiler fires.

The after-bulkhead is required so as to isolate leakage which may be
caused by the breaking of the stern tube, or by general vibration due to the
action of the propeller. Usually, it is placed near enough the stern to
prevent any loss of buoyancy consequent on the bilging of the after com-
partment being sufficient to endanger the vessel. Incidentally, it forms a
splendid stiffener at this part, an important consideration when the machinery
is placed aft.

Although no surveyor to the Board of Trade, under the existing regula-
tions, could refuse to grant a declaration of survey that the hull, even of a
passenger steam-vessel, whatever her length, was sufficient for her work, if
fitted with bulkheads equivalent to the foregoing, such an arrangement can
clearly be considered satisfactory only in small steamers. With increase in
size, additional bulkheads very soon become desirable, partly because of the
need of providing greater transverse strength, but as this may be met other-
wise, mainly because safety in the event of fire or bilging demands an
adequate sub-division of the holds.

Thus we find, taking Lloyd's latest Rules for example, that when steam-
vessels reach 285 feet in length, five bulkheads are necessary, the distance
between the collision and boiler room bulkheads being sub-divided. In vessels
of 2>35 f eet > tne a fter hold is in turn sub-divided, making six bulkheads, the
number of watertight bulkheads becoming seven, eight, nine, and ten, when
vessels reach lengths of 405, 470, 540, and 610 feet respectively.

Obviously, the question of sub-division is of first importance in purely
passenger vessels, no form of life-saving appliance being so efficient as a good
system of watertight bulkheads. Few first-class passenger steamers are there-
fore now built but can float safely with, say, any two* compartments in open
communication with the sea, while some have even a better sub-division,
and partly on this account, a few of these have been subsidised by the
Government to act as auxiliary cruisers in time of war.

A point of great importance in the fitting of bulkheads is that they
should extend well above the loadwater line ; otherwise the bilging of one
compartment may cause sufficient sinkage to submerge the tops of the bulk-
heads, in which case the water would find its way into all the compartments

* In the report of the Bulkhead Committee ot 1890, the highest class of sub-division
is given as that which would enable a vessel to float safely, in moderate weather, with any
two compartments in open communication with the sea.

1 63 SHIP CONSTRUCTION AND CALCULATIONS.

and thus sink the vessel. In general, bulkheads should extend to the top
deck of the main structure. In vessels with continuous superstructures, such
as an awning or shelter deck, the bulkheads (with the exception of the collision
bulkhead, which should extend to the awning or shelter deck) are usually
stopped at the deck below, i.e., the upper deck, in virtue of the greater free-
board and reserve buoyancy of this class. In the 'tween decks of these
vessels, a deep web frame or partial bulkhead is to be fitted on each side
immediately over the watertight bulkheads, or other efficient strengthening
must be provided.

CONSTRUCTION OF BULKHEADS.— Although an ordinary watertight
bulkhead may never be called upon to sustain the pressure due to a com-
partment on either side becoming filled, it must be constructed strong
enough to meet such an eventuality. It should, therefore, be built of plates
of substantial thickness, and be strongly stiffened. Lloyd's Rules require a
thickness of "26 of an inch in bulkheads having a depth from upper deck to
floors of from 8 to 12 feet, i.e., in the smallest vessels, and of '46 of an inch
in those in which the same depth is from 44 to 50 feet, i.e., in very large
vessels. The plates are fitted vertically or horizontally, are usually lapped at
edges and at end joints, and single-riveted, the rivets being spaced for water-
tight work, i.e., 4I diameters apart. At the points where the end joints come
on the edges, the two plates of the former are thinned down for the breadth
of the edges laps so as to obviate the fitting of slips.

In the stiffening of watertight bulkheads, the plan recommended by the
Bulkhead Committee of 1S90 is now usually followed, the stiffeners being
arranged generally in a vertical direction (see also stiffening of collision bulk-
heads). In the former Rules of Lloyd's Register, bulkhead stiffeners were
required to be arranged horizontally as well as vertically — a cross-bracing
arrangement which assured the strength of the bulkhead in a transverse as
well as a vertical direction, making it efficient to resist pressures tending to
force in the ship's sides, which a purely vertical arrangement of stiffeners is
not adapted to do. By the cross arrangement, too, the unsupported area is less
than by the vertical. Still, the advantages of an entirely vertical arrangement
of stiffeners are considerable. To begin with, as the depth of a bulkhead is
less than the width, stiffeners are shorter and therefore more efficient arranged
vertically than when arranged horizontally — the rigidity of girders varying in-
versely as the cubes of their lengths. Again, the pressure which a bulkhead
may be called upon to withstand is greatest at the bottom, and a range of
closely-pitched vertical stiffeners bracketed to the tank top are effectively placed
to resist this.

The spacing of stiffeners in ordinary watertight bulkheads should not
exceed 30 inches. In the case of a collision bulkhead, as a vessel's safety
may depend on this bulkhead's ability to withstand the dashing about of
masses of water admitted to the fore peak through collision, the spacing of
vertical stiffeners should not exceed 24 inches, and in this case there should
also be horizontal stiffeners consisting of bulb angles on the opposite side,

CONSTRUCTION OF BULKHEADS, 1 69

spaced 4 feet apart, bracketed to the ship's sides. As the horizontal stiffeners
are short, the vessel being narrow at this part, they add immensely to the
rigidity of the bulkhead.

Bulkheads which form the ends of oil compartments in vessels designed
to carry oil in bulk, or which form the ends of deep-water ballast tanks, should
be of extra strength, because, as well as fulfilling the main function of
ordinary bulkheads in affording sufficient structural strength, they must be
able to resist the pressure of the mass of fluid which the compartment con-
tains, the speed of the vessel being communicated to the fluid in a com-
partment through the bulkhead at its after-end ; also, as any compartment on
occasion may not be quite full, its bulkheads should be strong enough to
meet the very severe stresses which the dashing about of large masses of
fluid in a partly-filled tank would give rise to. Lloyd's Rules provide scant-
lings for the bulkheads of oil vessels.

Frequently, the edges of plates forming bulkheads are flanged to act as
stiffeners. This entails a vertical arrangement of somewhat narrow plates,
since the distance between stiffeners must not be more than 30 inches.
There is here a saving in riveting, and fewer parts require to be put
together; this system is therefore rather popular, especially as experiments have
shown the arrangement to be as strong as the ordinary one, and as mild
steel may be readily flanged cold. Lloyd's Rules require that when flanged
stiffeners are 12 inches or more in depth, intercostals are to be fitted between
the stiffeners and connected to the bulkhead plating and to a bar on the
face of the stiffeners (see fig. 159). These intercostals, which are to be spaced
not more than 10 feet apart, should greatly stiffen the bulkhead by prevent-
ing any tendency to trip on the part of the stiffeners.

In Lloyd's Tables the scantlings of the stifTeners are shown to vary with
the full depth of the bulkhead as governing the maximum pressure that
could come upon it. When the bulkhead is divided into zones by the
abutment of steel decks, the scantlings of the lower stiffeners, that is, those
between the tank top, or in single bottom vessels, the top of floors and
lowest laid deck, are governed by the full depth as fixing the intensity of
the pressure, and the length of the stiffener as fixing the load. 'Tween deck
stiffeners are, in the same way, governed by the distance from the top of the
bulkhead to the lower part of the 'tween decks, and by the length of the
stiffener. Stiffeners in way of holds and 'tween decks, except the upper
'tween decks, should be bracketed top and bottom. This follows from the
consideration that a uniformly loaded girder fixed at the ends is 50 per cent,
stronger and five times more rigid than one with free ends. Lloyd's Rules
permit bulkhead stifTeners, in small vessels, to be fitted without end brackets,
provided their scantlings be increased beyond the tabular requirements. In
oil vessels, which are generally built without inner bottoms in way of the oil
holds, the knee brackets at the lower ends of the bulkhead stiffeners should
be fitted between the floors to the shell.

At the edge every watertight bulkhead should have a strong connection

170

SHIP CONSTRUCTION AND CALCULATIONS.

to the shell-plating, inner bottom (where one is fitted), and deck-plating.
Double angles are frequently fitted to the shell-plating and inner bottom and
make a good job, but it is becoming increasingly common, particularly in
cargo vessels, to have instead large single bars double-riveted in both flanges •
the latter arrangement is cheaper and is probably not less strong. Lloyd's
Rules make provision for both methods. Reference has already been made

Fig. 159.

to the shell liners required in way ot outside st rakes, as compensation for the
closer spaced rivets — necessary for caulking — through the shell angles of water-
tight bulkheads. When the shell liners are not fitted, as with joggled plating
or framing, bracket knees between the shell-plating and the bulkhead in way
of outside strakes are necessary, except where the hold stringers are 5 feet
or less apart. The joints on one side only of a bulkhead require to be

CONSTRUCTION OF BULKHEADS. I7 1

caulked. Where hold stringers and keelsons pass through bulkheads, caulked
angle collars should be fitted on the watertight side, and to give a finished
appearance, plate collars, uncaulked, on the other side. Frequently, hold
stringers are stopped at the bulkheads, and the longitudinal strength is made
good by fitting substantial bracket plates connected to the bulkhead by angles,
and to the stringers by a riveted lap (see figs. 159 and 160).

The subordinate bulkheads of a ship, such as screens and casings, do
not call for a lengthened description. They are constructed of light plates
and bars, the former having single-riveted joints. They are not usually
watertight, and where perforated by beams, dust tightness is secured by
fitting plate collars. Where screens take the place of pillars, as in the case
of side bunker casings and centre-line bulkheads, additional rigidity is called
for, and is obtained by increasing the thickness of plating and making the
stiffeners of substantial size, the latter being fitted two frame spaces apart in
line with the beams and attached thereto. Where vessels have open floors,
the centre-line bulkhead is attached to the vertical plate of the centre keelson.
A centre-line bulkhead is usually stopped in way of the hatches, so as not
to interfere unduly with siowage, and, when required for grain cargoes, the
continuity of the division is made good by wood shifting-boards. Although
interrupted in this way, when properly built, a centre-line bulkhead is a splendid
vertical web, excellently adapted to resist longitudinal deflecting stresses.

When machinery casings in 'tween decks have to take the place of
quarter pillars, they must be strongly built, and the stiffeners should be
riveted to the beams.

DOORS IN WATERTIGHT BULKHEADS.— It is, of course, desirable
that watertight bulkheads should be intact, as their efficiency as subdivisions of
a hold is then at its highest. In some cases, however, doorways must be
cut in them. For instance, the need of a direct means of access from
the engine-room to the shaft tunnel, calls for a door in the watertight bulk-
head at the after-end of the engine-room, where the tunnel abuts upon it.
Again, in most cargo steamers, a reserve coal bunker lies immediately before
the forward boiler-room bulkhead, in which doors must be fitted so that the
coal may reach the stokehold floor. In special cases doors have been
fitted at the ceiling level in all the watertight bulkheads of a vessel, when
it has been desired to pass from hold to hold without going on deck.
Besides the foregoing, particularly in passenger vessels, doors are frequently
made in watertight bulkheads at the height of the 'tween decks, so that
passengers may readily get from place to place in the region devoted to
their accommodation.

In designing doors for watertight bulkheads it is necessary to remember
that one placed near the foot of a bulkhead would have to withstand con-
siderable pressure, if from any cause a compartment on either side of it
became flooded. The framing of the doorway and the door itself are
therefore made specially strong. Usually these parts are of cast iron of
substantial thickness. The door is made of wedge shape, as also the

172

SHIP CONSTRUCTION AND CALCULATIONS.

groove in which it works, any degree of tightness of the joint, which is a
metal to metal one, being thus obtainable. As in the case of bilging a
door would quickly become inaccessible, arrangements must be provided for
working them from a high level. Doors in engine and boiler-room bulk-

Fig. 160.

UrPERDECK.

SECTION

bulb angles
so'apart

CONNECTION OF STRINGERS TO BULKHEAD

331

SECTION CF

SIDE STRINGEff
heads are usually wrought from an upper platform ; doors in other bulkheads
are worked from the deck. Where doors open in a vertical direction (fig.
161) the apparatus for working them commonly consists of a vertical shaft
with a screw at one end working in a fixed nut in the door. Where they

DOORS IN WATERTIGHT BULKHEADS.

173

open in a horizontal direction, the vertical shaft is fitted at its lower end
with a small pinion wheel which works a fixed rack on the door.

Doors in bulkheads which give access into 'tween decks need not be
designed to withstand great water pressure. Usually, they consist of plates
hinged to the bulkhead and secured by snibs so fitted as to be readily
operated from either side of the bulkhead. The joint between the door
and the door frame on the bulkhead is made watertight by means of spun
yarn or rubber packing (fig. 162).

STEMS, STERNPOSTS, AND RUDDERS.— In merchant vessels the

SHAFT TOR OPERATING
DOOR

Fig. 161

BULKHEAD

DETAIL SECTION
OF DOOR FRAME

l t4t j I

-DOOR

COVER PLATE
FORMING GROOVE FOR DOOR

stem consists of a solid forged bar of iron or steel, or of rolled steel, of
. suitable breadth and thickness, and forms the fore-end of the hull. Nowadays,
stems are usually straight above the load-waterline with a slight rake— say,
two feet— forward, to minimise the effect of a collision, should this happen,
and to overcome the impression of falling aft at the head which a quite
vertical stem gives. The clipper stem, so common at one time, is now
seldom built on steamers ; in sailing ships, as it is a suitable construction
with a bowsprit, and also has a fine appearance, it is always found. When
associated with a hanging or bar keel, the stem becomes a continuation
of the same, being connected to it by a vertical scarph similar to that em-
ployed for uniting the lengths of the "keel bar. When the keel is of centre

T74

SHIP CONSTRUCTION AND CALCULATIONS*

through plate or side bar type, a modification of the ordinary vertical scarph
is adopted. By referring to fig. 163, it will be seen that the after-end of the
stem is slotted out to receive the ends of the centre girder and the two side
bars, which together make up the thickness of the keel* The total length of
this scarph should be about eighteen times the keel thickness, Or double the

Fig 162,

DETAIL SHEWING FASTENING

-RUBBER PACKING

BULKHEAD 35 PLATING

WEDGE 5*I J £* J £ TAPERINCTQ^'

DOOR 35 PLATINC

DETAIL AT HINGE

OVAL PINHOLE "
TO ALLOW DOOR TO CLOSE TIGHTLY

HORIZONTAL SECTION

ALTERNATIVE PLAN

BULKHEAD

^^^^^^ /P ; RUBBER

DOOR

length of scarph required for an ordinary bar keel, to allow a reasonable
distance between the terminating points of the flat side bars. There are
several ways of making a connection between a stem bar and a flat plate
keel. Fig. 164 shows one adopted by many builders. The lower part of
the stem is carried three or four feet on to the fore length of the keel,

STEMS.

175

and is securely riveted to intercostal plates, which in turn are riveted to the
floors. The lower ends of the frames in this vicinity extend below the top of
the stem bar to the line shown dotted in the figure, and the fore end of
the keel-plate is dished so as to come under the stem and yet fay against
the ship's side. The keel-plate may be said to end where it rises on to
the side of the stem (fig. 164), as in front of that point it becomes an
ordinary strake of shell-plating. The preceding is an efficient plan, and
obviates the necessity of tapering down and spreading out fanlike the after-
end of the stem — a more costly arrangement, but one which gives good work
and formerly frequently adopted. It will be observed from fig. 165, which
illustrates this method, that the keel-plate is dished round the after part of
the stem, and continued for the distance of a frame space, or two under it.
Thence, as in the previous case, the keel-plate is lifted on to the side of

Fig. 163.

END OF CONTINUOUS CENTRE
GIRDER

(NTERCOSTALS

ALS — ^

Vr^jsr-TT"

t \

TACK. RIVETS
KEEL SIDE BARS

STEM

E-

SCARPH ]

the stem and through riveted to it. Through riveting is also adopted at the
after-end of the stem, if practicable, otherwise tap riveting is resorted to.
The centre keelson-plate is carried intercostally for a few frame spaces for-
ward of the after-end of the stem and attached to a tongue formed on the
stem, as in the sketch, or in lieu of a tongue to bottom bars tap-riveted
to the stem.

As well as by means of the keel connection, the stem is thoroughly bound
with the structure by the main shell-plating. The strakes at their forward
ends are arranged to lap on each side of the stem, and rivets sufficient in
length to pass through all three thicknesses are employed (see fig. 166). It
will be noticed that the shell-plating is kept back 1-inch from the front of
the stem ; this is to protect the caulking. At least two rows of rivets are re-
quired to connect the shell-plating to the stem, and these should have the same

176

SHIP CONSTRUCTION AND CALCULATIONS.

spacing as the keel rivets, viz., 5 diameters, centre to centre. Below the load
waterline the stem should be maintained at full thickness, as it is there liable
to severe strains by grounding or collision ; above that point it is usually
reduced somewhat. In practice it is tapered to the top, where the sectional
area has three-quarters its maximum value.

To facilitate construction and reduce the cost of repairs, in the event of
damage to the stem, the latter is usually made in two parts with a scarph
at about the light waterline. Mention may here be made of the practice of

collision

8ULK

Fig. 164.
ELEVATION

/LINE OF FRAME HEELS

SECTION AT CD.

LOOKING An

SECTION AT A. B.

LOOKING F0RW°

NTfR COSTAL

using tack rivets in stem scarphs. They are fitted to join the parts together
for the purpose of erection and fairing, but they are a drawback when a
portion of the stem has to be removed, as plates on both sides of the stem
have to be taken off in order to punch them out, for which reason they
are frequently omitted.

STERNPOSTS.— The sternpost forms the after-end of the hull structure.
In sailing-ships and paddle steamers, and also in some twin-screw steamers,
it consists, like the stem, of a simple bar, with the addition of forged gudgeons
for hinging the rudder. In single-screw steamers, however, this part of the
ship becomes more complicated, for in addition to providing facilities for

STEMS AND STERNPOSTS.

176a

carrying the rudder, the propeller shaft, which leaves the hull at this point,
must be supported by it. Moreover, the strain caused by the continual
working of the shaft has to be counteracted, and this can only be done by
making p the post and its connections to the hull of ample strength. Fig.
167 shows the stern frame of an ordinary cargo steamer. The stem of this
vessel is 11 inches by 3 inches, and the increase in strength of sternpost
necessary, for the reasons given, is represented by the increase of the thickness

Fig. 165.
ELEVATION

tNDOFKCEL PLATE

Fig. 166.

of the propeller post, which is joined to the shell-plating, to 9 inches, the
breadth remaining n inches, while the rudder post, which is not called
upon to withstand such severe stresses, may be 9^ inches x 9 inches. Besides
this, as previously mentioned, the after lengths of the shell-plating, which come
upon the propeller post, are augmented in thickness above adjoining plates,
being usually of midship thickness, while the plates in way of the bossing
round the shaft are still further thickened. The shell-plating is attached to
the stern frame by two rows of rivets of large diameter, increased below

i>]6b

SHIP CONSTRUCTION AND CALCULATIONS.

the boss in vessels over 350 feet in length to three rows. The hull at-
tachment is further improved by securely connecting the upper arms — marked
A and B* in fig. 167— to floorplates, also by extending the arm G well
forward, and connecting it to the keel-plate and middle line keelson.

The size of the aperture is fixed by the diameter of the propeller, for
the efficient working of which ample clearance must be allowed. It is of

Fig. 167.

importance to keep the propeller as low down as possible so as to ensure
its always being under water, as when partly immersed, the efficiency is much
reduced. For this purpose the lower part in way of the aperture is reduced
in depth and increased in width, the sectional area being increased 15 per
cent, over that of the propeller post, as this part has frequently to with-

*Arm B is required by Lloyd's Rules in vessels whose longitudinal number is 16,000 and above.

STERN P9STS. 176^

stand severe grounding stresses. The main purpose of the after-post is
for hanging the rudder, for which the necessary braces or gudgeons are
provided, as shown. These should be spaced sufficiently close to properly
support the rudder. In Lloyd's Rules tabulated distances are provided
on the basis of the diameter of the rudder stock ; in the Rules of the
British Corporation, gudgeons are required to be spaced 4 feet apart in vessels
of 10 feet depth, and 5 feet 6 inches apart in vessels of 40 feet depth and
upwards, the spacing in- vessels between 10 feet and 40 feet depth being
found by interpolation. Gudgeons should have a depth equal to j^ of the
diameter of the rudder stock. These details of the sternpost are best con-
sidered in association with the rudder. When vessels are of large size it
becomes impracticable to make the stern frame in one piece. Moreover,
as the part outside the hull proper is most liable to damage, it facilitates
repairs and makes them less costly, if this portion can be easily discon-
nected from the remainder. We, therefore, usually find that stern frames in
large modern single-screw steamers are built up, as shown in fig. 168,
with scarphs as shown. The upper joint can be disconnected without dis-
turbing the main structure, while the lower one only interferes with the after-
length of the lowermost strake of shell-plating. These scarphs should have
a length equal to three times, and a breadth equal to i| times, the width
of the frames, and be secured by four rows of rivets.

It should be said that stern frames are built as just described, ue. y in
several pieces, only when they are made of cast steel ; but there seems no
good reason, except the extra expense and difficulty of forging the scarphs,
why, in ordinary simple cases, forged stern frames should not be so made,
considering the advantages accruing thereto.

A word may here be said in regard to the relative merits of cast steel
and forgings for stern frames, rudders, etc. The rules of the classification
bodies permit the use of cast steel for such items, subject to their with-
standing certain tests, and as castings are cheaper than forgings they are
populaf with some builders. But, from an owner's standpoint there are
objections to castings. They are, for instance, not as reliable as forgings,
for while flaws in the latter are rare, inherent weaknesses, acquired in the
processes of manufacture, frequently exist in stern frames and rudder castings,
and these, though not disclosed by the usual tests, are sure to manifest
themselves subsequently when the parts are in place and doing their work.
Again, a defect in a forged frame may frequently be effectively, quickly, and
cheaply repaired, but a serious flaw in a steel casting simply means its re-
newal, which, in . addition to considerable expense, may cause loss to the
owners in delaying the ship. It is only fair to state, however, that in recent
years there has been improvement in the manufacture of large steel castings.

Of course, where the forms of stern frames and rudder are complicated,
as in the case of some war vessels and large passenger liners, steel castings
are resorted to because forgings are quite impracticable.

ij6d

SHIP CONSTRUCTION AND CALCULATIONS.

So far, reference has been made exclusively to the stern frames of single-
screw steamers, but those of modern twin-screw vessels call for special
mention. In the simplest form, as in the case of small vessels, the stern

Fig. 168.

RUOOGR ARMS AT **
BETWEEN PINTLES

PLAN OF RUDDER ARM

frame proper is of the familiar L-shape fitted to saiiing-ships, the projecting
propeller shafts being supported by means of a A bracket on each side.
This form is illustrated in figs. 169, 170. In the first figure it will be

STERNPOSTS.

176*?

observed that the upper palm of the bracket is fitted directly on to the
shell, which is doubled in the vicinity, and the lower one is through-riveted

Fig. 169.

SECTION

LOOKING AFT

STRONG BEAM
AT PALM

DETAIL AT UPPER PALM

"CHECK FOR SHELL PLATING
CHECK FOR PALM

KEEL INCREASED IN DEPTH
IN WAY OF PALM

Fig. 170.

DETAIL AT UPPER PALM

ANCLE COLLAR

V /

000
000

0^
000

to the keel, .the latter _.being made deeper for the purpose at the place
required, a strong plate beam, connected to the sides by deep bracket

i 7 6/

SHIP CONSTRUCTION AND CALCULATIONS.

plates, being fitted across the ship in way of the palms, to give the neces-
sary rigidity at this part. In the second case, the upper palm is riveted to
a plate inside the ship, an angle collar being fitted round the strut where it
passes through the shell-plating, and the lower palm is riveted to a projection
forged or cast on the lower part of the stern frame.

Very often, in order to keep the lines of shafts near the middle line,
and thus minimise vibration as well as protect the propellers, the latter are
overlapped, a screw aperture thus becoming necessary. The usual arrange-
ment in such cases, when associated with a brackets, is as shown in fig.
171. The aperture must be of sufficient width in a fore-and-aft direction

Fig. 171.

to take both propellers; it need not, however, be so high as for a single
screw, the upper point of the propeller path being clear of the middle line.
In the special instance before us, the stern frame is designed in such a way
as to take the palms of the A brackets, the whole being riveted together.
Other arrangements might easily be devised, although that shown is very neat.
The different fore-and-aft positions of each propeller is arrived at by making
the shaft bossing longer on one side than on the other.

The A bracket system of supporting the propeller shafts, though simple,
is not suitable where high speeds have to be attained. Experiments have
shown that in such cases the projecting brackets cause a serious augmenta-
tion of resistance. It was found, for instance, in one case, that of a twin-

PROPELLER BRACKETS.

176^

screw vessel of fine form, the propeller shafts of which were encased in tubes
supported by two sets of struts, that the resistance caused by the tubes
amounted to 4J per -cent, and by each set of struts to about 10J per cent.
of the total hull resistance. Various attempts have been made to overcome
this . objection by giving a suitable shape to the arms, which from a more
or less circular section, in early vessels, became of a flattened oval shape
in those more recently built. The results obtained in this way were better,
but the resistance was still serious. The strut resistance being mainly due
to the disturbance of the stream lines, an attempt was latterly made to
eliminate this by bossing the form of the vessel round the shafts, right up

Fig. 172.

to the stern frame, thus allowing the streams an unbroken run aft.* This
plan, although somewhat costly, has otherwise proved most satisfactory, and
is now frequently adopted, particularly in fast vessels of large size.

As well as reducing resistance to speed, bossing the hull round twin-
screw shafts has an obvious advantage in that it adds much to the strength
at the after end. It also obviates the possibility of any lateral strain being
brought upon the shafts, as might happen where they are exposed.

" A model of the liner Kaiser Wilhelm der Grosse, when tried in the experimental tank
at Bremerhaven, was found to have 12 per cent, more resistance with propeller brackets than
when fitted with shaft bossing. — Engineering, 9th October, 1908.

176/1

SHIP CONSTRUCTION AND CALCULATIONS.

When properly constructed the bossing becomes an integral part of the
hull structure. Fig. 172 shows in section the method of constructing the
fin — as it is sometimes called — where the distance from the ship's side to
the line of shafting is considerable. As will be seen, each main frame is
carried down in the ordinary way, and on the reverse side a bar suitably
shaped is fitted and made to overlap the main frame for some distance
above and below the points at which it leaves the normal frame line. The

Fig. 173.

S1DL ELEVATION
STERNPOST

PLAN

frame-work is strengthened by web-plates, as shown. All the frames in the
bossing need not be built in this way. For a considerable distance the
eccentricity in form may be met by bossing out the main frame, the object
being to obtain the required shape and strength as economically as possible.
For the purpose of forming bearings for the shafts and a termination
to the bossing, a special steel casting, sometimes described as a spectacle
frame, is fitted Fig. 173 shows this frame associated with an ordinary

STERNPOSTS. I 7 6/

propeller frame having an aperture — a fairly common arrangement. In the
figure the spectacle frame forms part of the propeller post ; frequently it is
a distinct casting bolted to the propeller post, which is complete without it.

RUDDERS. — The rudder is that part of a vessel which controls the
direction of her movements when -afloat and in motion. As the axis about
which a vessel turns is in the vicinity of amidships, and as the rudder
takes the * deflecting force, obviously the best position for it is at either end
of the vessel. The after-end is most convenient for the purpose, and, with
a few exceptions, it is always placed there.

In most mercantile vessels the rudder is hinged about an axis at its
forward end (figs. 168, 174); in war vessels, and in some few merchant ships,
what is termed a balanced rudder is fitted, having the a*is so placed that
about a third of the area lies before it. The obvious advantage of the
latter type consists in the ease with which it can be put over to port or star-
board. It has a disadvantage, however, in being somewhat costly, a sufficient
reason to debar its adoption in ordinary cargo vessels, especially, 'as owing
to their low speed, a rudder of common type can be operated by a steer-
ing gear of moderate power.

Fig. 168 shows the rudder frame of a modern cargo steamer of large
size. It is seen to consist of a vertical main frame or stock with arms at
right angles to it, the latter being spaced close enough to afford sufficient
support to a heavy plate which gives the contour of the rudder. The arms,
which are forged or cast with the rudder frame, are arranged on alternate
sides of the plate, as shown in the sketch. The rivets attaching the arms
to the rudder-plate should be of large size, and the arms kept back a little
from the outside edge of the plate to protect them from being torn off.
The rudder is attached to the stern frame by means of bolts or pintles,
which ship into gudgeons on the after-part of the stern frame. These
gudgeons are forged or cast solid with the stern frame, and are afterwards
bored out at the ship as required, care being taken to keep their centres
in line so that the rudder may have a true axis. Formerly, the rudder
pintles were also forged on the rudder frame, but are now usually portable
bolts, as in the illustration given.

Fig. 1 74 shows a style of rudder frequently fitted in modern vessels.
It has a circular stock and arms that are separate forgings fitted one at
each pintle. This is about double the spacing of the previous case, to
allow for which the wider spaced arms are made relatively heavier. In
fitting the parts together the post is turned in way of the arms, which are
shrunk on, a key being fitted to prevent the arms turning. Usually a
groove is cut in the back of the stock for the rudder-plate to fit into, the
stresses on it being thus communicated directly to the stock and the rivets
in the arms to some extent relieved.

The weight of the rudder, in most cases, is taken by the bottom gudgeon
of the sternpost. Fig. 175 shows this arrangement in detail. The socket for
the bottom pintle -is not continued through the gudgeon as with the others-,

176;

SHIP CONSTRUCTION AND CALCULATIONS.

but sufficient housing is allowed to prevent any danger of accidentally un-
shipping. In the present instance, the depth of the socket is 4 inches. To
minimise friction, the bottom of the pintle is rounded, and a suitable bearing
provided by fitting a hemispherical steel disc into the gudgeon socket.
Experience with this style of bearing has not shown it to be completely

Fig. 174.

SINGLE PLATE RUDDER

ARMS AT PINTLES

satisfactory. The weight of the rudder soon produces wearing, which is usually
uneven, the friction then becoming greater than if no disc were used. The
hole from the bottom of the socket to the heel of the post is to enable
the disc to be easily removed.

Rudder pintles are all alike except the bottom one, which is some-
what shorter than the others, and the "lock"' pintle, to which we shall
refer presently. The part of each pintle which fits into the rudder frame is

RUDDERS.

176/$

tapered from bottom to top, to prevent its being knocked out. On the head
of the pintle a large nut is fitted, which secures it in position, any slacken-
ing tendency being guarded against by a steel pin which is driven through
the pintle immediately over the nut, as shown.

To prevent accidental unshipment of the rudder, a locking arrangement
must be devised. A simple plan is to make one of the pintles — preferably
the top one — with a bottom collar (fig. 176).

Another point of importance for the satisfactory working of the rudder
is to provide a means of limiting the turning angle, which, in ordinary cases,
should not exceed 35 to 40 degrees. Referring to fig. 177, which shows a
common design of stopper, it will be seeri that the movement of the rudder

Fig. 175.

beyond a certain inclination is checked by widening out one of the gudgeons
on the sternpost and altering the shape of the rudder stock in the vicinity,
so that each surface may bear solidly on the other at the required angle.
In a very large vessel two such stoppers would be needed, and they should
be fitted so as to distribute the pressure equally over the sternpost. Stoppers
must also be fitted on deck, the rudder movement being here controlled by
stopping the quadrant or tiller arm. Where a good brake is fitted to the
tiller, or the quadrant is geared on to the steam steering engine, no deck
stops are necessary, the control being sufficient without them.

The foregoing is a description of a rudder such as is fitted in an ordinary
cargo vessel, and it will be observed that only bare essentials are provided
for. Where an owner does not object to extra expense in order to obtain
greater efficiency, refinements are introduced. For instance, it is advantageous

T76/

SHIP CONSTRUCTION AND CALCULATIONS.

to bush the gudgeons with brass or lignum-vilae (fig. 178), and more so
to also line the pintles with brass or gun-metal (fig. 179). By these means
the rudder is made to work more smoothly, and as the parts, when worn, can
be renewed with little trouble or expense, a high standard of efficiency is easily
maintained. The objection to carrying the weight of a rudder on the
bottom gudgeon has already been referred to. This has sometimes been
overcome by causing the rudder to bear on several or on all the gudgeons, cir-
cular discs or washers of white metal being inserted between the rudder lugs
and the gudgeons for this purpose. Another plan is to fit solid washers,
cone-shaped at bottom, into each gudgeon, with pintles having tapered points
to suit. The weight of the rudder is thus distributed over all the gudgeons,
and there can be little or no side movement of the rudder. By both
these arrangements, of course, more power will be required to turn the
rudder than when it is supported on a footstep bearing only, but this is

Fig. 176.

Fig. 177.

no drawback where there is an efficient steam steering gear. Occasionally
rudders are fitted which do not bear on the gudgeons, the weight being taken
by a thrust block inside the vessel, usually fitted at the level of the transom
floor. With balanced rudders this is the invariable plan, the bottom pintle,
where there is one, serving merely as a guide. The fitting of an internal
bearing to a rudder of ordinary type adds to the cost, but it has the ad-
vantage of accessibility, an important consideration when dealing with working
parts.

When rudders increase greatly in size and weight, it becomes necessary
to devise a simple means of shipping and unshipping them without disturbing
the steering gear and inboard stuffing boxes. It is customary, in such cases,
to fit a coupling just under the counter, and this is found to answer the pur-
pose admirably. Horizontal couplings, as illustrated by figs. 168, 174 ancj
180, are common, although others of a vertical type are sometimes fitted (figs.
181 and 181a). With such an arrangement, to unship a rudder it is only

RUDDERS.

t 7 6m

necessary to unscrew the pintle nuts, thus allowing the pintles to drop out,
and to disconnect the rudder coupling. By means of block and tackle, the
rudder may then be easily moved out of its usual position.

Fig. 178.

i i

Fig. 179.

Fig. 180.

Nowadays, the rudder proper is usually formed by a single heavy plate
as previously described ; another plan, once universal, and still sometimes
followed, is to design the frame to the desired contour, as illustrated in fig.

i76«

SHIP CONSTRUCTION AND CALCULATIONS.

182. Each side of this frame is covered by thin plating, through-riveted, the
space thus enclosed being filled in solid with wood or cement. This style is

Fig. 181.

M'LACHLAN'S vertical coupling

.-9 0IA'

Fig. 181a.

v-H

'WEDGEW0OD5 SCARPHED

coyPLirsci

BOLTS g lDlA'

not so strong as that of the single plate; it is also more liable to decay
through corrosion, as the inside surfaces of the rudder-plating are obviously

RUDDERS.

1760

Fig. 182.

inaccessible for- cleaning. These were the chief reasons of its abandon-
ment in ordinary vessels in favour of the single-plate type. In special
cases, such as yachts, it is still retained for
its finer appearance.

The sizes of the various parts of a rudder
are governed by the area and shape of the
latter, and the speed of the vessel. Knowing
these particulars, the twisting moment can be
determined and the requisite diameter for the
head of the rudder stock calculated. The
aggregate sectional area of the^arms support-
ing the single plate depends to some extent
on the bending moment to be sustained, but
it should be increased beyond this requirement
to allow for shocks from the sea, to which the
rudder in stormy weather may be subjected.
The sizes of the pintles should also be suffi-
cient to withstand these shocks and provide
for wear and tear, considerable at these parts.
The strength of the coupling joint must be
equal to that of the stock. This entails flanges
of considerable thickness and a sufficient num-
ber of coupling bolts, the moment of whose
aggregate strength about the rudder axis
should be equal to the twisting moment, and,
therefore, to the torsional strength of the
rudder stock.

These are the principles which must be
followed in making detailed calculations. Of
course, if a ship is to be built to Lloyd's
Rules, such calculations, on the part of
the builder at all events, are unnecessary,
as detailed dimensions of rudders are pro-
vided in carefully compiled tables. In these
the diameter of rudder stock is given^ for vari-
ous speeds under numbers which represent
the product of the total area of the rudder
in square feet abaft the centre line of the
pintles, and the distance in feet of the
centre of gravity of this area abaft the same
line.

As previously remarked, rudders are sometimes fitted at the fore ends of
vessels, such, for instance, as have to navigate channels too confined to turn
in. These rudders are usually designed to come inside the line of the
stem, and to follow the shape of the vessel, being thus more or less

SECTION THROUGH A B.

176/

SHIP CONSTRUCTION AND CALCULATIONS.

buoyant (fig. 183). The rudder stock is carried to the weather-deck and
worked by a simple hand-gear. As a bow rudder is mainly for emergency
purposes, when not in use it is locked in a fore-and-aft position by means
of a strong bolt.

Fig. 183.

PLAN OF TOP OF RUDDER

CHAPTER VII.

Equilibrium of Floating: Bodies: Metacentric

Stability.

FROM our considerations in Chapters I. and II., we know something of
the forces in operation when, as depicted in fig. 184, a vessel is
floating freely and at rest in still water. We know, for instance —

i . That the total upward forces, or buoyancy, mus t equal the total
downward forces or weight ;

2. That the resultant of the downward forces acts through <?, the centre
of gravity of the weights, and the resultant of the upward forces through B,
the centre of gravity of the displaced fluid, already denned as the centre
of buoyancy.

It is now necessary to note that these two equal and opposite resultant

*£

Fig. 184.

IL_

forces must act in the same vertical line, for, if the lines of action did not
coincide, a turning moment would be in operation to disturb the equilibrium!

Now, suppose an external force to act upon the vessel and cause her
to heel over, as shown in fig. 185. No weights have been added, there-
fore the displacement is unchanged, and the volume lifted out of the water
on one side must be counterbalanced by the volume immersed on the
other ; that is, the wedges W\S W and L X S L are equal.

As the immersed body is now altered in form, the centre of buoyancy

is no longer at B but takes up some new position B x ; and as there has

been no change in the disposition of the weights, the centre of gravity G

is not altered in position. The two equal resultant forces act clown through

M 177

i 7 8

SHIP CONSTRUCTION AND CALCULATIONS.

G, and up through B l respectively, their lines of action having a perpen-
dicular distance GZ between them, as drawn in the figure. The turning
moment acting on the vessel obviously tends to restore her to the original

Fig. 185.

position, and she is therefore said to be in stable equilibrium. Next,
suppose the centre of gravity to be raised from the position in fig. 184
say, by pumping out a ballast tank, and by putting a quantity of cargo

Fig. 186.

into the 'tween decks in order to keep the displacement the same, or by
some other means. First, let G become exactly coincident with M (see
fig. 186). As before, the weight will act downwards through G s and the

Fig. 187.

buoyancy upwards in the line B 2 M. The forces will therefore act in op-
posite directions in the same vertical line, and being equal in magnitude
will neutralise each other. In this case there will be no lever tending to

DEFINITION OF TRANSVERSE METACENTRE. 1 79

heel the vessel, which will not tend to depart from its inclined position.
The condition is said to be one of neutral equilibrium. Now, let G be
raised above M. A glance at fig. 187 will show what will happen if the
vessel be inclined as before. The two resultant forces will act in different
lines, causing a heeling moment to be in operation on the vessel. There
is, however, a very important difference between this heeling moment and
that existing when G was below M> the tendency being now, not to right
the vessel, but to incline her further from the initial position. With G
above M, therefore, the vessel, when in the upright position, is in unstable
equilibrium.

We thus see that the relation between the points G and M in a
floating vessel entirely determines the nature of her equilibrium. M is called
the metacentre from its being the meta or limit beyond which the centre
of gravity G must not rise, if a condition of stability is to be maintained.
It may be defined as follows ; —

Definition of Transverse Metacentre. — If a vessel be floating up-
right at rest and in equilibrium, at a certaifi draughty and be then inclined
through a very small angle \ the point in which the vertical line through the
new centre of buoyancy intersects the middle line of the ship } is called the
transverse metacentre at that draught. For every draught there is, in
ordinary vessels, a different position of metacentre. The point also changes
with every inclination from the upright. It is usual, however, and sufficiently
correct for practical purposes, to assume it as fixed for inclinations up to
10 or 11 degrees. This is important, as within these limits, if we know the
distance G M, we can determine the vessel's righting power, since —

Moment of Statical Stability in foot-l ,,, , _ ..

tons at any angle 6 \ - W x B Z - Hf x Q M x Sm I,

W, the displacement, being given in tons, and G Z or G M in feet.

This is known as metacentric stability, G M being called the metacentric
height It must be borne in mind that this method applies only up to the
angles above given ; beyond these it is unreliable, as M changes rapidly in
position, and G M has no longer its initial value, which is the only one that
is used by the metacentric method. Further on we shall see, when considering
actual curves of stability, that in many cases considerable error would be
involved, even at moderate angles, by using the above formula for calculating
the moment of stability.

A knowledge of a vessel's metacentric height is, however, useful for many
purposes. It is an excellent guide, for instance, for determining whether or
not a vessel may be safely shifted in harbour, or whether ballast tanks may
be run up, or, in the case of a vessel carrying oil in bulk, how the loading
of cargo should be proceeded with. In conducting the first of these opera-
tions, there need be no inclination from the upright exceeding that for which
the moment of stability may be written —

W x GM x Sin ft
so that in order to shift the vessel with confidence, it is only necessary to

l8o SHIP CONSTRUCTION AND CALCULATIONS.

make sure that the value of G M is sufficient. In the two last operations
G M should be great enough to allow for the reduction in its value due to
the presence of free liquid in the vessel. We shall return to this point again.
Besides the foregoing, if the vessel be of known type, the metacentric height
will furnish a good basis from which to predict the probable nature of
her stability at large angles of inclination.

The great importance of the points G and M will now be manifest, and
a shipmaster ought to know for every condition of lading of his vessel in
which she may have to put to sea, what GM or metacentric height he has
available.

In considering these two centres, the influences controlling the position
of each should be carefully noted. Obviously, G is fixed by the distribution
of the weights, and we shall show presently how it may be determined in
any given case. The point M, however, is not affected by the weight dis-
tribution, but only by the underwater volume of the vessel, and by the shape
of the waterplane. This appears from the formula that gives the height of
the point relatively to the centre of buoyancy, which may be written —
Height of transverse metacentre "i R M I
above centre of buoyancy / ' I/ 1

where / is the moment of inertia of the waterplane about its middle line
as axis, and V the volume of displacement.

The numerator of the right-hand member of this equation may be ex-
plained in a popular way, as follows : — Imagine the area of the whole waterplane
to be divided into an infinite number of parts, and the distances of the centres
of these elements from the middle line ascertained; then, if each of these
small areas be multiplied by the square of its distance from the axis, and the
sum of the products be taken, the result will be the moment of inertia required.

Although it involves some calculation to obtain the above moment of
inertia in the case of an ordinary-shaped vessel, owing to the varying nature
of the boundary line of the waterplane, it may be quickly obtained for any
figure of simple form such as a square, a circle, or a triangle, as established
formulas are then available. We have a case in point in a floating box-shaped
vessel. Here the outline of the waterplane is a rectangle, and the moment of

L R 3

inertia of this figure about the major axis is — , where L is the length of

the vessel, and 8 the breadth.

Applying the formula for the height of the transverse metacentre above
the centre of buoyancy we have —

L x B*

V LxBxD i2 0'

D being the mean draught. If the actual dimensions of the vessel be — length,
150 feet; breadth, 30 feet; draught, 15 feet; then —

12 x 15

CALCULATION OF BM. 131

Almost as simple a case occurs when the vessel is of constant triangular
section with the apex down. The waterplane is, as before, a rectangle, so
that the expression for / is unchanged, but the displacement is obviously only
half the previous value, and we now have —

L x B*

! B 2

BM =

LxBxD 6D

We therefore note that a floating vessel of this form has its transverse
metacentre at twice the height above the centre of buoyancy of another having
a rectangular section, the extreme dimensions in each case being the same.
Moreover, in a vessel of triangular section, the centre of buoyancy is at a
greater height above the base line than in the other case, so that the abso-
lute height of the metacentre is, on this account, still further increased. Now,
the forms of the 'midship sections of ordinary ship-shaped bodies lie between
the two extreme cases just considered, and, neglecting for the moment the
influence of tapered lines, the general effect of change of design upon the
position of the transverse metacentre may be grasped. It is important to
note in the above formula that B M is independent of the length, while the
breadth appears in the second power. This shows the influence of breadth
on stability, and explains why broad shallow vessels have always high meta-
centres.

A unique case occurs where the vessel is a floating cylinder with its
axis horizontal. In ordinary vessels the metacentre, as we have seen, may
be considered as a fixed point only for one draught. In this case, however,
the vertical through the centre of buoyancy will intersect the middle line
at the same point at all draughts. This will be apparent, if we consider
that, since the immersed section is part of a circle, a normal to any water-
line through its middle point will pass through the centre of buoyancy,
and intersect the middle line of the vessel at the centre of section. Thus,
for a vessel of cylindrical section, there is only one position for the trans-
verse metacentre.

In applying the formula B M = -rj to ship-shaped bodies, the work, as

already stated, mainly consists in obtaining the value of /. In actual calcula-
tion it is usual to divide the waterplane into an even number of equal parts,
suitable for the application of Simpson's First Rule, to treat the cubes of the
ordinates, measured at the points of division, as ordinates of a new curve,
and find the area of the latter in square feet, two-thirds of the quantity
so obtained being the moment of inertia of the waterplane about the middle
line. To obtain the value of the height of the transverse metacentre above
the centre of buoyancy, this moment of inertia, as we have seen, must be
divided by the volume of the ship's displacement in cubic feet up to the
waterplane or draught considered. As practical examples of the foregoing, and

l82

SHIP CONSTRUCTION AND CALCULATIONS.

in order to impress the method upon us, we shall calculate the value of
B M in two particular cargo vessels, both of modern type. The first is a
small deadweight carrier of full co-efficient, having the following dimensions: —
length, 275 feet; extreme breadth, 39 feet, 6 inches; moulded depth, 20 feet,
3 inches; the load draught is 18 feet, 9 inches, and the displacement
4535 tons. We shall deal with the vessel when in this condition. The
work is given in the table below. In the first and second columns we
have the numbers of the half ordinates of the load-waterplane, reckoning
from the after end, and their breadths as measured at the points of division ;
in the third column are tabulated the cubes of these half ordinates, and in
the fourth and fifth, Simpson's Multipliers and the products of these multi-
pliers with the cubes, respectively.

No. of
J. Ordinates.

J Ordinates.

(£ Ordinates). n

Simpson's
Multipliers.

Functions of
(i Ordinates). 3

[

IO'2

Io6l

2122

16-3

4331

il-

6496

19*0

6859

-743 6

i9"5

7415

14830

i9"5

7415

29660

i9'S

7415

14830

i9*5

7415

29660

i9'S

7415

14830

i9'3

7189

28756

14*9

33°3

4962

ioi

8-5

614

1228

—

174810

Height of transverse metacentre) n „ 174810x27*5x2

> = d m = = 0*75 teet

above centre of buoyancy

3 X 3* 4535 x 35

It will be observed that the figure 3 appears twice in the denomina-
tor of the expression for the value of B M, once as required by Simpson's
Rule, and once for the moment of inertia calculation.

The other vessel chosen for illustration is of somewhat finer form, and
much larger. Her dimensions are — length, 469 feet, 4 inches ; breadth, ex-
treme, 56 feet; depth, moulded, 34 feet, 10 inches. This vessel at a draught
of 27 feet, 6 inches, has a displacement of 15,814 tons. We shall find the
value of B M at this draught, arranging the work in tabular form, as in the
previous case —

APPROXIMATE CALCULATION OF 1!M.

183

No. of
£ Ordinat.es.

4 Ordinates.

Q Ordinates). 3

Simpson's
Multipliers.

Functions of
(^ Ordinates). 3

13*0

2197

4394

21'5

27'5

9938
20797

14907
83188

27-9
27-9

21717
21717

2
4

43434
86868

27*9
27-9

21717

2
4

43434
86868

9
10

27*9
27*0
18-8

21717

19683
6644

2
4

43434

78732

9966

10J

IO*2

I06l

2122

1 1

—

■ —

497347

As before-

BM _ 497347 x 46-93 x 2 = ^ ^

3 x 3 x 15814 x 35

APPROXIMATE METHODS FOR FINDING BM.—U in the two pre-
ceding examples the vessels were treated as of rectangular form of the same
extreme dimensions, and the rule applied, we should get for the small vessel —

39'5 x 39*5

and for the large one-

12 x 18*58
S 6 x 56

= 7 feet ;

= 9*6 feet.

12 x 27^25

The draught in each case is reduced by the depth of a flat keel. These
results are sufficiently near the actual values to suggest the possibility of
framing an approximate rule for readily obtaining the height of the metacentre
for ordinary vessels, but employing the formula as for box-shaped vessels,
with factors or co-efficients introduced to make up the differences between
the types. Now, we know that the moment of inertia of a waterplane

B 3 L

of rectangular shape about the middle line is , where B is the full

breadth and L the length ; this may also be written —

/ = Ci x B 3 x /., where C x = — , or -083.

For ship-shaped load waterplanes of moderately fine form, C 2 * will vary

from '05 to '055 ; and where they are of full form, from '06 to '065.

Thus, we are able to arrive at a ready expression giving the value of the

numerator in the formula for B M. The denominator may be similarly

* These co-efficients are from Sir Wm. Whyte's Manual of Naval Architecture, to which
the reader is referred for figures applying specially to war vessels.

1^4 SHIP CONSTRUCTION AND CALCULATIONS.

treated, as the volume of a ship's displacement may always be written —
V = t x B x D x G 2 , where C 2 is a co-efficient varying with the form. Using
these approximations, we get —

bM I/' C t LttD- 2 X D~ H D-

For many classes of merchant vessels k = '09, while in vessels of full form
it may become as low as *o8, or even less ; in fine ships, such as yachts,
k may rise to '15.

As a test, let us apply the approximate formula to the two examples
for which we have made detailed calculations. For the small vessel, which

has full ends, k = ~ = — ^ = *o8o6 ; and therefore —
^2 7 8 7

BM = -0806 x 39 ' 5 Q X ^ 9 ' 5 = 6-77 feet.
18-58

In the larger vessel, the ends ot the load waterplane and the underwater
body are both finer, and making due allowance —

and BM = -0818 x ^ — *LL = 9 - 40 feet.
27-25

Values of height of metacentre above centre of buoyancy thus obtained are
therefore seen to approach the actual figures very closely.

In order to fix the position of the metacentre in the vessel, it is necessary
to know the height of the centre of buoyancy. In approximate calculations,
for vessels of ordinary form, this value may be taken as varying between ^
and 2°o of the mean moulded draught, measured downwards from the water-
line, the latter figure being used for full vessels. For detailed calculations,
the position of the centre of buoyancy, as obtained from correct drawings oi
the vessel, must, of course, be employed.

While it is most important to know the position of a vessel's transverse
metacentre when floating at her load draught, it is frequently necessary to
know it for other draughts. For some classes of vessels the launching condi-
tion is a critical one, and the amount of GM available then should be known.
Cases are on record of vessels capsizing through deficient stability, while
being launched.

Another important condition for which the metacentric height should be
known is that called "light-ship," which means that the vessel is complete,
including machinery, but is without cargo or bunker coal. This condition
forms an excellent basis from which to calculate the value of GM for the
vessel when laden with any kind of cargo.

Still another condition calling for special consideration is that when in
ballast. Modern cargo vessels frequently perform voyages in ballast trim,
and calculations should be made to find the disposition of ballast which will
give a value of GM, ensuring the good behaviour of the vessel at sea.

DIAGRAM OF METACENTRES.

185

Thus we have, including the loaded one, four conditions for which it is
essential to know the positions of the transverse metacentre and the centre of
gravity. To enable us to find the former quickly at any draught, a diagram is
constructed showing the change in the position of M with change in draught.
(The centre of gravity must be dealt with specially, as we shall show afterwards).
The curve of metacentres is usually plotted on a diagram, such as fig. 188, on
which the curve of centres of buoyancy is also drawn — the distance between
the two curves at any point being the value of B M at the corresponding
draught. In plotting the curve of metacentres, the procedure is the same as
for the curve of centres of buoyancy, which we may assume to be already

Fig. 188.

plotted. That is to say, referring to fig. 188, the height of M for various
draughts is calculated and spotted off on A B. Then each of these points
M } , M^ M s , etc., is translated out horizontally, a distance equal to that between
the load draught and the draught to which it refers, and a curve is drawn
through them. To complete the diagram, a line A A y at 45° to the vertical
A B is drawn from the point A, where the load waterplane intersects A B. To
obtain, now, from such a diagram, the value of B M at any draught, BE say,
it is only necessary to draw a horizontal line at that draught to intersect the
line AA x aX some point E lt and to draw through the latter point a vertical line
to the curves of buoyancy and metacentres at B and M . It will be clear,
after a little consideration, that B M is the height of metacentre above the
centre of buoyancy corresponding to the draught BE.

i86

SHIP CONSTRUCTION AND CALCULATIONS.

In fig. 189 are shown, in one diagram, curves of metacentres con-
structed in this way for prismatic floating bodies having cross sections
of rectangular, triangular, and circular form, marked respectively R /?, T 7",
C 0. The curve marked applys to a cargo vessel of ordinary form
with full lines. This diagram is very instructive. It will be noticed that
the locus of M for a vessel of triangular section is a straight line which
falls as the draught diminishes — a characteristic to be found in the dia-
grams of fine vessels as they approach very light draughts, the immersed
volume being then more or less triangular in form. The locus for a cir-
cular section, as might be expected, is a horizontal straight line, M coin-
ciding with the centre of the section for all draughts. The curve for the

Fig. 189.

vessel of box form resembles that for the ordinary ship in being convex to

the base; that is to say, the position of M at first falls, as the vessel

02
lightens. For a box-shaped vessel, B M = — -, so that, since B is constant,

the value of B M continually increases as D diminishes. The convex shape
of the curve of metacentres is, therefore, entirely due to the fact that, at
first, the centre of buoyancy falls more quickly than the value of BM in-
creases. This peculiarity in the curves of metacentres of vessels of full
form should be carefully noted, as we see that if the position of the centre
of gravity be assumed unchanged, while a vessel rises from the load
draught to another somewhat less ; the initial stability will be reduced,
although the " freeboard," or height of the deck above the waterplane, will
be increased.

TO FIND POSITION OF CENTRE OF GRAVITY. 187

METHODS OF FINDING THE POSITION OF THE CENTRE OF
GRAVITY. — A knowledge of the position of the transverse metacentre at
any draught, as provided by such diagrams as the above, is of itself of no
value whatever in predicting a vessel's initial stability. For example, we may
have two similar vessels with identical curves of metacentres, and yet at the
load draught one may have excessive initial stability, and the other be
unstable. As stated already, it is the relation between the positions of M
and G which is of paramount importance. In the similar vessels just
referred to, the condition as to stability has been entirely influenced by the
position of the latter point. In the stable vessel the heavy items have
been placed low down ; in the other, the opposite has been the case.
This shows how much the behaviour of a vessel at sea depends on those
who have charge of her stowage.

Fortunately, G may be determined very easily by means of an experi-
ment, and being thus known for a given condition, the effect of a new
disposition of cargo on the initial stability may be closely estimated. As
well as by experiment, G may be found directly by calculation. This is
the method employed by the naval architect in the preliminary stages of a
ship's design in arranging the positions of the fixed weights. In warships,
yachts, and other vessels, which sail at practically constant displacements,
the estimate for the centre of gravity must be very carefully made, since
any defect in the stability on completion cannot, without great expense, be
corrected. In freight-carrying vessels the stowage of cargo, as we have
seen, greatly influences the final position of 6, and its "light-ship" position
is therefore not of the same importance.

We cannot, in this work, elaborate in all its details the calculation
method of finding G. It is, however, perfectly simple in principle, consist-
ing, in fact, of a huge moment calculation, in which every item of a ship's
weight, including her cargo, is multiplied by its distance from two datum
lines at right angles in the middle-line plane, the centre of gravity being fixed
by the values obtained when the sum of each of these systems of moments is
divided by the total weight of the vessel. The datum lines are taken in the
middle-line plane, as obviously, since both sides of the ship are alike, the
centre of gravity must lie in that plane. The experimental method, which
we now proceed to explain, briefly consists in heeling the vessel by moving
a weight across the deck, observing the consequent effect upon the ship's
centre of gravity, and thence deducing the value of GM. From this, since
the position of M is known, the height of G may then be determined.

In carrying out an experiment it is important to see, in the first place,
that the vessel is floating quite freely, /.<?., not aground at any point, or
unduly hampered by neighbouring vessels, or too tightly moored to the
quay. Indeed, it would be better if moorings could be unloosed altogether.
The condition of the vessel should next be noted. If she be "light," the
holds, ballast tanks, and bunkers, should be empty. If some coal still
remain in the bunkers, it should be trimmed level so that its weight and

IO» SHIP CONSTRUCTION AND CALCULATIONS.

position of centre of gravity may be determined. The weights and positions
of all items which may still require to go on board to complete the vessel,
such as deck machinery, small boats, etc., must also be noted. A correction
is made afterwards to allow for the effect on the final position of centre of
gravity by the removal or addition of these weights. If the vessel should
be in loaded trim, the ballast tanks will probably be empty, but they should
be carefully sounded, and any loose water pumped out. Loose water in
ballast tanks or holds is particularly detrimental to such an experiment.
The apparatus may now be got ready. This consists of the heeling weight,
a plumb line and bob, and a straight edge. For a vessel of fair size, the
weight should not be less than about ten tons, so as to ensure a definite
inclination when the weight is moved across the deck. Successful experi-
ments have been carried out with a lighter heeling weight, when the dis-

Fig. 190.

tance moved has been considerable, as, of course, the heeling moment, which
consists of the product of the weight into the distance moved, may be made
up of a heavy weight into a short distance, or vice versa.

The plumb line is usually hung in the middle-line plane of the vessel.
A convenient place, when the holds are empty, is at a hatchway, the line
being suspended at the upper-deck coaming, and the movements of the bob
weight marked on a straight edge arranged for the purpose on top of the
ceiling. With a loaded vessel this will, of course, not be possible, but a
mast-stay, if screwed up tightly, will do quite well to suspend the bob weight
from, and a straight edge on which to record the movements of the latter
could be fitted in a suitable position near the deck. The method of conduct-
ing the actual experiment may now be described. In the first place, half the
heeling weight is arranged on each side of the upper deck, at a place
allowing an unobstructed passage across the ship. The vessel is in the upright

TO FIND POSITION OF CENTRE OF GRAVITY. 1 89

position, and the point N where the plumb bob crosses the straight edge
is carefully marked (fig. 190).

The weight on one side is then moved through a distance a feet across
the ship, as shown, causing the plumb line to move out of the centre and
take up a position R K. Now, moving the heeling weight from one side
of the ship to the other causes the centre of gravity of the whole structure
to move in the same direction through a distance given by the equation —

G G\= — 777-, where W is the heeling weight in tons, a the distance it is

moved in feet, and W the total displacement in tons. The point G\ is
clearly the centre of gravity of the vessel in the inclined condition. Since
there is equilibrium, the upward line of action of the resultant buoyant
force must be in the vertical GJH ; and M being the intersection of this
vertical line with the middle line, it is, by definition, the metacentre.
From inspection, the triangles N R K and G M G l are similar, and therefore,

GM JV_R

GG,~N K*
The inclination being very small, the distance NR may be taken as the
length of the plumb line. We therefore get —

GM = length of^umbline x0fly

This is the metacentric height (uncorrected), and all that remains to be
done to obtain the position of the centre of gravity above the base line,
is to deduct this distance from the height of the transverse metacentre
above the same line at this draft, as measured from the diagram of meta-
centres. Corrections are afterwards made to allow for the removal of the
inclining weights, and for the addition and deduction of other weights, if
such be necessary to bring the vessel into the desired condition.

It is the custom with certain shipbuilders to heel their vessels for the
position of the centre of gravity when in the first two of the four conditions
previously mentioned, viz., the " launching " and " light " conditions. With
other shipbuilders the "light-ship" condition is the only one dealt with.
The information obtained for the light condition is frequently supplied to
the shipmaster, to be used as a basis for making estimates of the position
of the centre of gravity when in any actual service condition, such as when
in ballast, or when fully loaded. In these estimates it is, of course,
necessary to have a strict account of the weights of the various items of
cargo or ballast put on board, with the positions of their centres above
the base, or any other datum line, and to combine the whole in a moment
calculation. For ballast conditions, particularly where the ballast consists
of water in fixed compartments, and for loaded conditions with homo-
geneous cargoes, this method is quite 'reliable ; for loaded conditions with
miscellaneous cargoes, however, it is not so satisfactory, as it is to be
feared that the care required to ascertain the weight and the centre of

190 SHIP CONSTRUCTION AND CALCULATIONS.

gravity of every individual item of the cargo, would not always be exercised •
and without such care, the calculated position of the ship's centre of gravity
might be very wide of the mark. To make sure of the condition of his
vessel with regard to initial stability, when fully loaded with a mixed cargo,
a shipmaster can always resort to a special heeling experiment, which we
have seen to be simple in character and absolutely reliable. As a practical
example of such an experiment, let us take the case of the smaller of the
two vessels (see page 182), whose nietacentre at the load draught we have
found to be 675 feet above the centre of buoyancy, as recorded particulars
of a heeling experiment carried out on her when fully loaded are available.
At the time of the experiment the vessel, including the heeling weights, had
a displacement of 4535 tons. The plumb line was hung from a stay and
was 23 feet, 6 inches long. The inclining weights, arranged in two lots of
five tons, were placed one lot on each side of the deck, at equal distances
from the* centre line, the distance between their centres being 33 feet. The
deflection N K of the plumb line caused by moving one portion of the
weight across the deck from port to starboard, was found to be 6\ inches.
As a check, the weight transferred across the deck was replaced in its old
position, and an observation taken of the plumb line. It should, of
course, have returned to the middle line, but scarcely did so. The other
portion of the inclining weight was next shifted from starboard to port, and
the resulting deflection of the plumb line noted ; it was 5! inches. In
the calculation, the mean of the observations was taken, viz., 6 inches.
Now —

1 4535 "* J

23*5
and, therefore, GM = —— x '0363 =171 feet.

Assuming the positions of the nietacentre and the centre of buoyancy
to be given, we have the following : — ■

Height of nietacentre above centre of buoyancy = 675 feet.

Height of centre of buoyancy above base line = 875 feet.

Height of nietacentre above base line = 15-5 feet.

Distance of centre of gravity below nietacentre = 171 feet.

Height of centre of gravity above base line — 1379 feet.

The inclining weights were removed from the ship, and this was the only
correction necessary. These weights being situated above the ship's centre of
gravity, the effect of their removal was to lower the latter point. Calling
the displacement of the vessel including the inclining weights W, the inclin-
ing weights W, the uncorrected height of the centre of gravity above the
base /;, the height of the centre of gravity of the inclining weights above

CALCULATION OF GM, 191

the base /; and taking moments about the base (the weights being in tons,
and heights in feet), we have —

Corrected height of centre) W x h - W x /
of gravity above base ) W - w '

4535 x 1 319 - 21 x 10 . A

= H*2 2-iZ 2 = 1377 feet.

There was a slight fall in the position of the centre of buoyancy due to
the reduction in draught, but also a slight increase in the value of B M, the
height of M above the base remaining as before. The corrected metacentric
height thus became 15*5 1377 = 173 feet.

To estimate the change in the position of the centre of gravity due to
raising or lowering weights already on board, or removing them from the
vessel altogether, is now a very simple matter. Let us take a specific case : —
Assuming 250 tons of cargo are to be discharged from the bridge 'tween decks
of the above vessel, at a certain port, find to what extent the centre of
gravity and metacentric height will be affected. Taking moments about the
base line, we have —

New height of centre of gravity^ 4525 x 1377 — 250 x 26*5
above base line )~ 45 2 5 - 2 5° ^

To get the corrected G M, we must allow for the fact that the position of
M is altered by the change in displacement and in the form of the water-
plane. Assuming no change of trim to take place —

Weight of cargo removed

Change in draught

Tons per inch of immersion
= -5- = 1 1 '36 inches, or "95 feet.

The new draught is, therefore, 1875 -0*95 = 17*8 feet. From the curve of
metacentres the corresponding height of transverse metacentre above the base
line is 15*8 feet, and thus we have —

GM = 15-8 - 13-02 = 278 feet.
As a further example, take the case of the large cargo steamer previously
dealt with (see page 182). This vessel's weight, when in the "light" condition,
i.e., ready for sea, but with no coal or cargo aboard, is 5134 tons, the centre
of gravity being 20 feet, 6 inches above the top of keel. Assuming the dis-
placement scale and diagram of metacentres to be available, let us find the
GM when laden with 10,680 tons of cargo and bunker coal distributed as
follows: — 6420 tons of cargo in lower holds, 3320 tons in shelter 'tween
decks, and 940 tons of coal in bunkers. Tabulating our data, we have —

Light weight of ship = 5134 tons.
Deadweight = 10680 tons.

Total displacement — 15814 tons.

192

SHIP CONSTRUCTION AND CALCULATIONS.

Turning to the displacement scale we find the draught corresponding to
this displacement to be 27 feet 6 J inches to bottom of keel. The transverse
metacentre above the base line at this draught (from diagram) is 23-5 feet.

To obtain the position of the centre of gravity, we must make a
moment calculation, as follows : —

Weight in
Tons.

0. of G.
above base.

Moments.

Light Ship

Cargo in holds

Cargo in shelter 'tween decks

Coal in bunkers

5*34
6420

33 2 °
940

20"5

15*5
4i'o

24'0

IO5247

995 10

136120

22560

15814

363437

Height of centre of gravity) 363437
above base line J

= 22*98 feet,

15814
so that GM = 23*5 - 22*98 = '52 feet.

This value is less than would be considered safe, unless the corresponding
curve of stability were particularly favourable. If we haven't got this informa-
tion, it will probably be considered desirable to remove some of the cargo
from the shelter 'tween decks into the main 'tween decks, so as to lower
the centre of gravity 4 or 5 inches. If 10 feet be the distance through
which such cargo may be lowered, the quantity affected is given by the
equation —

Weight of cargo to be lowered x 10 = Displacement x fall in centre of gravity.
Substituting values we obtain —

Weight to be lowered = — — =522 tons.

The new position of centre of gravity will be 22*98 - "33 = 22-65 ^ eet above
the base line, and the value of GM '52 + '33 ="85 feet.

The burning out of the bunker coal has often an important effect on the
stability. Let us find what it would be in the present case. Assuming the
weight of coal to be 940 tons, and its centre of gravity 24 feet above the
base, the effect of burning out the coal in the present instance would be to
lower the centre of gravity. Taking moments about the base line —

New height of "^ 15814 x 22*65 ~~ 94° x 2 4
centre of gravity/ 15814-940 ' *>

The fall in draught of water would be —

94Q

= IT*

I7f inches,

53 being the tons per inch of immersion at the load draught. The new
draught would therefore be 27' 6£" 1' 5!" = 26' o$". From the diagram,

APPROXIMATE CALCULATION OF INITIAL STABILITY. 193

the height of metacentre above base at this draught is 23-25 feet; so that
under the assumed conditions —

GM = 23*25 - 22*56 = '69 feet.
The burning out of the coal would thus reduce the initial stability.

As there would be draught to spare, it might be considered desirable
to run up some water ballast in order to bring G M to about its previous
value. Let sufficient water be supposed admitted to lower the vessel's
centre of gravity 3 inches. Then, assuming the centre of gravity of the
ballast to be 2 feet above the base, and taking moments about that line,
we have, representing the weight of the ballast by B —

(14874 + B) 22'3i - B x 2 = 14874 x 22*56

, . , n 14874 x 22-56- 14874 x 22-31

from which B = ' 3 — — =183 tons.

20*31 J

The added ballast would increase the draught — - = 3J inches, and from
the diagram we find that the metacentre would rise half-an-incb, therefore-
New value of G M = "69 + '25 4- '04 = -98 feet.

APPROXIMATE METHOD OF CALCULATING THE EFFECT IN THE
INITIAL STABILITY DUE TO ADDING OR REMOVING WEIGHTS OF
MODERATE AMOUNT. — In order to make estimates of a vessel's meta-
centric height, or initial stability, like the foregoing, considerable data must
be available. In many instances a shipmaster may not have this informa-
tion. By an approximate rule* he may, however, still find the effect on
the initial stability of raising or lowering, adding or removing, a moderate
weight, provided he knows its amount and the distance of its centre of
gravity from the load-waterplane. If to be the weight in tons, and h its
distance in feet from the load-waterplane, by this rule, the stability at an
inclination of 6 degrees will be affected to the extent w x h x Sin 6 foot
tons, the correction being a decrease if w is added at some point above
the waterplane, or removed from some point below it, and an increase if
the conditions be the opposite of these. Thus, the effect of running in
183 tons of ballast on the initial stability of the vessel referred to above,
will, by this approximate method, be to increase it by the amount —

(26-04 - 2 ) x ^3 x Sin 6 foot tons = 4399 Sin Q foot tons.

By the exact method, viz., Righting Moment =W x GM x Sin 0, the increase
is the difference between the stability after adding the ballast and that
existing before, that is —

(15057 x -98 - 14874 x '69) Sin foot tons = 4492 Sin Q foot tons,
so that the approximation is a good one.

* See a paper by Sir Wm. Whyte in volume XIX. of the Transactions of the Institution of
Naval Architects.

194 SHIP CONSTRUCTION AND CALCULATIONS.

As a further example, suppose 200 tons of deck cargo to be taken on
board at n feet above the load-waterline. The effect will be to reduce
the initial stability by the amount (200 x n) Sin foot tons, which
at 10 degrees is 2200 x '1736 = 381-9 foot tons. If the cargo were re-
moved instead, the initial stability would be increased by the same amount.
This method only gives reliable results when the addition or removal of
the weights causes no appreciable change in the form of the load-line.

SAFE MINIMUM 6 A/. —The question of a minimum value of G M has
been the subject of much debate and difference of opinion. Those who
have favoured a large value have been confronted with the fact that great
stiffness conduces to bad behaviour at sea, as will appear when we come
to the subject of rolling. On the other hand, a very small value indicates
a crank vessel, and may mean, although not necessarily so, an altogether
unsafe one. The only secure manner of dealing with vessels in this respect
is to compare them with others whose performances at sea are known, and
to adopt values of metacentric heights thus suggested.

In designing the various classes of warships, the value of the metacentric
height must be carefully considered from the point of view of what ma}
be expected from each vessel on active service. This is specially necessary,
'since G M cannot be readily altered after the completion of the vessel, the
weights being then fixed and the displacement more or less constant. If the
only consideration is to obtain a steady gun platform, G M should be small,
as a minimum of rolling motion at sea is thereby assured. If, however,
other considerations intervene, such as a liability to lose a portion of the
inertia of the load-waterplane when in action, to which some war vessels,
owing to their design, are subject, the steady gun platform must be sacri-
ficed and a sufficiently large initial value of G M provided to meet all
eventualities.

It is beyond our province to fully discuss the subject as affecting
warships, but coming to the case of trading merchant vessels, there seems
to be a consensus of opinion in favour of limiting the minimum value
of G M in steamers of about medium size to one foot when filled with
a homogeneous cargo, which just brings them to the load-waterline. Cases
are on record of vessels which have given a good account of themselves
with smaller metacentric heights, when loaded as above. In one oft-quoted
instance, the GM was as low as '6 feet, yet the vessel proved herself in
every way a good sea-boat. The natural feeling, however, is to have a margin
on the side of safety, and this is considered to be provided in ocean-going
steamers when the metacentric height has the minimum value given above.

For sailing-ships, of course; a much higher value of G M is requisite, in
order that they may not be unduly heeled when under canvas. The best
authorities give 3 to 3^- feet as a minimum value, and where a homo-
geneous cargo will not admit of this, ballast should be carried.

QUESTIONS ON CHAPTER VII. 195

QUESTIONS ON CHAPTER VII.

1. A vessel is floating at rest in still water; discuss the changes in the character of
the equilibrium as the centre of gravity is raised.

2. Define the transverse metacentre ; write down the formula for the height of the
transverse metacentre above the centre of buoyancy, and state its numerical value in the
case of a rectangular vessel 38 feet broad, floating on even keel at a draught of 20 feet.

Ans. — 6 'Oi feet.

3. What is metacentric stability? A vessel of 4000 tons displacement has a metacentric
height of 18 inches; calculate the stability in foot tons at an inclination of 10 degrees.

Ans. — 1041*6 foot tons.

4. The equidistant ordinates of a vessel's load-waterplane, measured on one side of the
middle-line at intervals of 16 feet, are — *2, 6-8, 9-6, ico, I0'2 io*o, 9*9, %•%, and 1 - 8 feet,
and half ordinates introduced at the ends are 3 '8, and 7 '3 feet, respectively. Find the
height of the transverse metacentre above the centre of buoyancy, the displacement being 530
tons. Ans. — 3 '46 feet.

5. A prism of circular section floats in still water with its axis horizontal. Show that
the metacentre is at the centre of the section for all draughts.

6. Two prisms of rectangular and triangular section, respectively, float on even keel
at the same draught; they are also the same breadth at the waterline. Show that the height
of metacentre above the centre of buoyancy in the first case is half that in the second. If
the breadth be 35 feet and the draught iS feet, what are the values in the two cases?

Ans. — 5-67 feet; 11*34 f eet -

7. A raft is supported by, and rigidly attached to, two rectangular pontoons placed
parallel to each other and 6 feet apart from centre to centre. Each pontoon is 4 feet wide
and 2 feet deep, and floats half immersed when the raft is laden. Calculate the meta-
centric height, assuming the centre of gravity of raft and lading to be 3 feet above the
waterline. Ans, — 6*83 feet.

8. Explain an approximate method of calculating the height of transverse metacentre
above the centre of buoyancy, and work out the numerical value in the case of & full-formed
cargo steamer of 48 feet beam and 25 feet draught. Ans. — 7*36 feet.

9. How is a. metacentric diagram constructed, and what are its uses ? Construct such
a diagram for a homogeneous rectangular prism afloat in still water, assuming it to be 30
feet broad and 20 feet deep.

10. The displacement of ^ vessel is 400 tons ; the transverse metacentre is 5f feet above
the centre of buoyancy, and the centre of gravity 3 feet above the centre of buoyancy. If
12 tons be moved 8 feet across the deck, find the inclination of the vessel.

Ans. — 5 degrees.

n. Explain how you would find the position of the centre of gravity of a ship—

(1). By calculation.
(2). By experiment.

12. The centre of gravity of a certain cargo vessel of 4500 tons displacement is found
to be 16 feet above the base; the following weights are then added, viz., 600 tons at 10
feet, 400 tons at 15 feet, and 500 tons at 11 feet above the base; the following weights
are removed, viz., 600 tons from 2 feet, and 100 tons from 14 feet above the base.
Calculate the new height of centre of gravity.

Ans. — 16'39 feet above the base.

19^ SHIP CONSTRUCTION AND CALCULATIONS.

13. How would you estimate the change in initial stability of a vessel due to raising or
lowering, adding or removing, a weight of moderate amount?

(1). Accurately.
(2). Approximately.

14. Estimate approximately the change in stability at an inclination of 10 degrees, due
to running in 150 tons of ballast at 2 feet above the base, and discharging 100 tons of
cargo from 30 feet above the base; the load draught to begin with being 22 feet.

Ans. — Stability increased 660 foot tons approximately.

CHAPTER VIII.
Trim.

IN the previous chapter we examined the condition of vessels when heeled
through small angles in a transverse direction ; in the present one we
propose to deal with longitudinal inclinations — that is, with the subject
of trim.

The vessel in fig. 191 is supposed to be heeled to a very small angle
in a fore-and-aft direction, by transferring a weight W tons from forward to aft
through a distance a feet. W^ and W L represent, respectively, the lines of

flotation before and after the transference. The draught is increased at the
stern and decreased at the stem, as shown. The sum of the distances L Z. : ,
and W W v is called the change of trim, and the finding of this for any
proposed condition of lading constitutes the trim problem. We shall en-
deavour to explain two methods of solution — one that ordinarily given in
text books \ the other not so well known.

In fig. 191, the point of intersection S of the new and the old water-
lines is at about the middle of the length. It actually coincides with the
centre of gravity of the waterplane W Y L^ which, in most cases, is a little aft
of the middle of the length ; however, it is sufficiently correct to assume

197

I9S SHIP CONSTRUCTION AND CALCULATIONS*.

SLj and S W x to be equal, and this simplifies the work somewhat. Re-
ferring to the figure, obviously —

Change of trim = W W x + L L x =■ (W.S + S Q tan Q.

= length of load line x tan Q (i).

Now the movement of the weight w causes the centre of gravity of
the vessel to be drawn aft through a distance G G x given by the equation

G G x = — r», — (W being the displacement in tons), and the resultants of the

vertical forces of weight and buoyancy, when the ship is once more in equi-
librium, act through the point G v This is indicated in fig. 191, as also the
previous line of vertical forces through G, corresponding to the condition before
the shifting of IV. Their point of intersection m is called the longitudinal
metacentre, and the distance Gm the longitudinal metacentric height, the
latter being of considerable importance in trim problems, as we shall see
presently. The angle between the two lines is clearly Q, the inclination
between the water lines, and therefore —

GG X = Gm tan 0,
or Gm tan = ^

w a

so that tan =

W x Gm

Change of trim
Length of load-line

Calling the length of load-line /., and equating these two values of tan Q, we

From (1) tan Q =

ing the length
obtain the relation

Change of trim w a , -.

I = WxGm {2h

which is the ordinary formula for change of trim due to any given shift of
weights already on board.

One or two simple examples will illustrate the application of this formula.
Take a vessel 200 feet long of 2000 tons displacement, in which the quantity
Gm is 190 feet, and find the effect on the flotation due to shifting 50 tons
aft through 80 feet. Transposing a little, and introducing the quantities
given, we have —

50 x So x 200

Change of trim — = ^'1 feet, say 2^ inches.

b 2000 x 190 ' J D

The draught forward will therefore be reduced, approximately, 12 J- inches,
and the draught aft increased by the same amount.

Again, if it were found that the propeller tips in this vessel were show-
ing 9 inches above the water when in the original condition, the minimum
weight w required to be shifted, say, 120 feet aft, so as just to immerse
the screw, would be —

i'ij x 2000 x 190

W = = 23*715 tons.

200 x 120 ° ' J

TRIM. 199

In dealing with questions like the foregoing, and indeed, in working out

any trim problem, it frequently saves time to find at the outset the moment

to alter trim one inch. To do this it is only necessary to substitute in

(2) 1 inch (or T \ foot, since the units are in feet) for "change of trim,"

and after transposing, Avrite down the equation giving the moment required.

Thus,

i-i W x Gm c L

Moment to alter trim 1 inch — w x a = —r— — ~ foot tons.

L x 12

In the preceding example —

2000 x 190

W x a = = 158"? foot tons,

200 x 12 ° ° '

and using this figure we get the same results as before more simply. Thus,
for the effect of shifting 50 tons aft through 80 feet we have —

-, .... Trimming moment 50 x 80

Change of trim in inches = tt — — ~rf — — : r— - = - — s — = 2*:.

b Moment to alter trim 1 inch 158*3 J

In the second question, as the change of trim necessary to submerge
the propeller tips is 18 inches, there must be a total trimming moment of
18 x 158*3 = 2849*4 foot tons, and as we know that «/, the weight, may be
moved 120 feet aft, obviously —

2849*4

W = = 2V7S tons -

1 20

The moment to alter trim one inch is thus seen to be very useful, and
in order to have it ready to hand for all possible conditions as to draught,
it is frequently found convenient to plot a curve of its values with varia-
tion in draught. In the calculations it is usual to employ B m instead of
Gm {see fig. 191) as, of course, the exact position of the centre of gravity at
the various draughts is unknown, and B m, as we shall see presently, may be
readily found. The difference between G m and B m, however, is usually small,
and for practical purposes the trim moments thus found are sufficiently ac-
curate. The plotting of the diagram is a simple matter, and as an exercise,
the reader should draw one for any case for which he may have the data.

LONGITUDINAL METACENTRE.— The only term in equation (2) call-
ing for special explanation is G 17), already described as the longitudinal
metacentric height — m being the longitudinal metacentre. The point m is
obviously analogous to M the transverse metacentre, with which we are already
familiar. In fact, the definition of M given on page 179 will also apply
to m, if for transverse the word longitudinal be substituted. The points
M and m have similar functions in respect to stability ; but vessels have
enormous righting power in a longitudinal direction^ and detailed calculations
of longitudinal stability are therefore unnecessary. The principal use of m
is found, as illustrated above, in dealing with questions of trim.

CALCULATION OF B m (fig 1 . 191).— As in the case of the transverse

200

SHIP CONSTRUCTION AND CALCULATIONS.

metacentre, the height of the longitudinal metacentre is first found above
the centre of buoyancy. The same formula is used, viz. : —

Bm = -n-i

but here / is the moment of inertia of the waterplane with respect to a
transverse axis through its centre of gravity. We shall see presently, that for
ordinary vessels the calculation of / is rather more laborious than in the
previous case ; for box-shaped or other vessels of simple form, this is, however,
not so. Take, for example, a box-shaped vessel of length /, breadth 6,
and draught d. Fig. 192 shows the waterplane, which is, of course, a
rectangle; xx, drawn at right angles to the middle line, is the axis of the

moment of inertia-

l about x x =

Fig. 192.
X

Since the volume of displacement is Ibd —

i 3 b _P_
12 / b d~ i2d'

Bm =

If it be given that / = 150 feet, and d = 15 feet,
then

ICO X KO

Bm = - j2 — — 7- = 125 feet.
12 x 15 J

If this vessel were of constant circular section, with its longitudinal axis
in the waterplane, the numerator of the expression giving Bm would be un-
changed, but the volume of displacement would be less. In such a case, we
should have —

Simplifying and substituting values

lb 2

•7854

r 5°

I50

6 x 30 x '7854

159*1 feet.

CALCULATION OF Bm.

201

Coming to ship forms, we find that the varying nature of the outline
of the waterplane and of the form of the underwater body introduces com-
plexity into the calculation. No simple formula is available for the moment
of inertia of the waterplane. To find this quantity it is usual to divide the
waterplane by ordinates in number suitable for the application of Simpson's
Rule, to calculate the moment of inertia of elementary strips of area at each
of these ordinates about some chosen axis, to treat these moments as ordinates
of a new curve, and calculate the area of the latter. This area, subject
to a further correction to be explained presently, gives the moment of
inertia required. The foregoing may, perhaps, be better understood by a
simple graphic explanation. In fig. 193, ABO represents a half of a load
waterplane of a vessel. x X is the axis about which the moment of inertia
is to be calculated, and is usually taken in the vicinity of the mid-length.

Fig, 193.

The ordinates 6 b 6 2 , 6 3 , etc., are numbered from aft, and the common
interval between them is h. Calling the breadth of each little strip a, we
may write —
Moment of inertia of elementary strip at b\ about x X = x (4A) 2 a — 0.

6 2 „ = 6 2 x (3//)* a = gb 2 h 2 a.

b 3 „ - 6 3 x (2hfa = 4b 3 h 2 a,
b 4 „ = 6 4 x {hfa = 6 4 A 3 a.
„ » b 5 „ = b 5 x 0. a = 0.

In the same way, moments of inertia of strips of area on the other side
of XX may be found. At the points of division on the middle line AC, the
the moments of inertia of the strips are erected as rectangles, the base in
each case being the small breadth a, and the ordinates, the quantities 0,
963 h% 4& 3 /? 2 , etc. Fair curves drawn through the tops of these rectangles
enclose areas as shown, which, added together, represent the moment of inertia
of the plane about the chosen axis.

202

SHIP CONSTRUCTION AND CALCULATIONS.

Now, the axis of the moment of inertia must contain the centre of
gravity of the waterplane area. If XX is not so placed, which generally
would be the case, a correction must be made. This is done by means
of the formula explained on page 63, viz.,

/, = / + Ak 2
where / is the moment of inertia of the plane about an axis through
the centre of gravity, I x its moment of inertia about any parallel axis X X y
h the distance between these axes, and A the area of the waterplane.
Applying this formula to the above case, it is necessary to find the dis-
tance k t which is obtained if the position of the centre ot gravity of the
plane is known. We have already seen how to find this latter point, and,
having obtained 7 h the value of the required quantity may be at once
written down.

As a numerical example, take the 470 feet vessel for which we have
calculated the transverse B M. The table below exhibits the full work, and
scarcely calls for explanation. It may be mentioned, however, that columns
4 and 6 are introduced to determine the area of the waterplane, and the
position of its centre of gravity from the assumed axis at ordinate No. 5 ;
also that, as each ordinate of the moment of inertia diagrams (fig. 193) in-
volves the square of the interval between the ordinates, the finding of the
areas introduces the cube of the interval in the expression for the moment
of inertia, as shown below.

No. of
Ordinates.

Ordinate.

S.M.

Functions of
Ordinates.

Levers for
Moments.

Functions for

Moments.

Levers
for
M.I

Functions of

Moment of

Inertia.

I 3 -0

2 6 'OO

II7*00

5 2 6-50

2i'5

32-25

I29'00

516*00

2 7 '5

I lO'OO

330-00

99COO

27-9

SS'So

ni'6o

223*20

4
5
6

27-9
27*9
27-9

4
2

11 i"6o
iii'6o

ni'6o

IIl6o
III - 6o

799*20

1 n'6o

27-9

55-80

111*60

223-20

2 7'0

108*00

324-00

972-00

iS"9

28-35

113-40

453'6o

IQ"2

20"40

91*80

413-10

—

715-60

752-40

799.20

4540.8

excess forward = 46*80

Centre of gravity of L.W.P. forward of No. 5 ordinate = 7~~^ = 3'o6 feet

715*60

Area of L.W.P. - '—^ ?-^A = 22388 square f ee ^

CALCULATION OF Bin, 203

4S40'S X (46 "Q^) 3 X 2

M.I. about axis through No. 5 ordinate = — — — 312890090.

M.I. about axis through C.G. of L.W.P. = 312890090- (22388 x (3'o6) 2 )

= 312680458.

The displacement of the vessel is 15814 tons;

I ^126804^8
so that, Bm = -77 = „ Q = 565 feet-

V 15814x35 3 *

For easy reference, values of B m, such as the above, are usually cal-
culated for various draughts and plotted in a diagram from which, given a
draught, the corresponding B ffi may be read off. Of course, for accurate
trim calculations, G m and not B m is required ; the distance between B
and G should therefore be deducted from the calculated distance B m, G
being usually above B. In the above case, for instance, the centre of
gravity is 6 feet above the centre of buoyancy, and, consequently —

Gm = 565 - 6 = 559 feet.

Fig. 194.

i=u_

The foregoing principles will be more clearly understood by the follow-
ing trim calculations for a few actual cases : —

Example 1. — Suppose 150 tons of cargo, having its centre 80 feet before
the centre of gravity of the load-waterplane, is to be discharged from the
above vessel ; what will be the new draughts forward and aft, it being given
that the vessel to begin with is on even keel at 27 feet 6 inches?

In questions such as this, and in those involving additions of cargo, it

is usual to assume, in the first place, that the removed or added weight

has its centre in the vertical plane, containing the centre of gravity of the

layer of volume which rises out of or sinks into the water. Under this

assumed condition, a vessel will obviously rise or sink through a parallel

distance. For, in fig. 194, if b be the centre of gravity of the layer through

which a vessel rises by the removal of a weight w/, and a be the distance

of b from the vertical through fl, the original centre of buoyancy of the

vessel, then —

W x a — Moment tending to depress bow,

the effect of the removal of the layer of displacement being to cause the
centre of buoyancy of the vessel to move aft : and if, as assumed, the
weight before removal had its centre in the vertical containing 6, another
moment due to removing w will also be in operation, in this case tending

204 SHIP CONSTRUCTION AND CALCULATIONS.

to raise the bow. These two moments are equal and neutralise each other,
and thus the vessel rises to the waterplane W1L1 without changing trim.
On the other hand, if the vessel be floating at the waterplane W x L b and the
weight w be added with its centre of gravity in the same vertical as 6, the
moment due to the increased buoyancy will be w x a, and will tend to
raise the bow, while that due to the added weight, being of equal amount
but of opposite sense, will tend to depress the bow ; so that the final effect
will be to sink the vessel to the line W L without change of trim.

It should be noted that, where the weights to be added or removed are
moderate, 6 may be taken as in the vertical containing the centre of gravity
of the waterplane.

Applying these principles to the working out of our question, it is
assumed that the 150 tons of cargo is immediately over the centre of gravity
of the waterplane, the effect of its removal therefore being to cause the
vessel to rise through a parallel distance —

150 150 .

,-« ■ — r — t~- = — = = 2f inches, nearly.

Ions per inch of immersion 53-3 4 J

The reader already knows how to find the tons per inch at a given water-
plane.

We next take account of the fact that the cargo is 80 feet forward of
the assumed position. A little consideration will show that the removal of
150 tons from a point 80 feet before the centre of gravity of the water-
plane, has the same trimming effect as the addition of the same weight
at 80 feet abaft that point. By removing the weight from its true posi-
tion, we therefore get —

Moment trimming vessel by stern — 150 x 80 = 12,000 foot tons.

Now,

u . . , W x Gm .
Moment to alter trim 1 men = — -. foot tons,

= 1570 foot tons,

12000 .

so that, Trim by stern = — — — = 7-f inches, nearly.

L x 12

i5 8l 4 x 559
470 x 12

2000
i57o

The final draughts will be —

Forward, 27' 6" - 2|" - 3 J" =26' n-|" ;
Aft, 27' 6" - 2 f + 3 f" = 27' 7 f .

We have assumed the change of trim to be divided equally at stem
and stern. This is not quite correct. Allowance should be made for the fact
that S (see fig. 191), the point in which the water-lines intersect, which coin-
cides with the centre of gravity of the load-waterplane, is not usually at the
mid-length, and the change of trim, forward and aft, should be allotted
according to the proportion —

LL X S±
WW X 8W

TRIM EXAMPLES. 205

In small changes of trim, however, it is not usual to proceed to this re-
finement, as the difference is inappreciable ; in the present case, it is less
than a quarter-of-an-inch.

Example 2.— A vessel 360 feet long, 48 feet broad, and drawing 26
feet aft and 20 feet forward, has to cross a bar which will only admit of
a draught of 25 feet. She has a fore-peak tank of 160 tons capacity.
Show by calculation whether the filling of this tank will trim the vessel
sufficiently to enable her to pass over the bar. The centre of gravity of
the water ballast is 166 feet forward of the centre of gravity of the load-
waterplane, the tons per inch of immersion is 33, and the moment to alter
trim one inch, 700 foot tons.

From the information given, we may write : —

Sinkage, assuming ballast immediately over^ 160
the centre of ffravitv of load-waternlane J = 77 = 4 ' 84 ' ^ 5 inches.

the centre of gravity of load-waterplane j 33
Trim by head due to actual position oi\ 166x160

= ^8 inches.

ballast / 700 °

Assuming waterplanes to cross at mid-length —

New draught forward =20' o'' + 5" + 1' 7'' = 22' o" >
New draught aft — 26' o" + 5" - 1 7'' — 24/ 10";

we thus see the vessel may, with care, be safely navigated across the bar.
In the two previous cases, the weights causing the change of trim are
small in comparison with the total displacements ; had they been large, it
would have been incorrect to assume b of the parallel layer to be in the
vertical containing the centre of gravity of the original waterplane. Its true
position is obviously somewhere in the line joining the centres of gravity
of the two planes enclosing the layer; do is this line in fig. 194. If the
areas A, A h of the planes WL and W\L h be known, the position of b may
be determined from the relation —

be A;

Very little error is involved if 6 be taken as the mid-point of tfe, and
in most cases this is done.

Another point to be borne in mind is, that in dealing with large
changes of trim, the plane of flotation, after movement of the weights, may
have become so altered in shape as to materially affect the value of the
moment of inertia, and, therefore, of the metacentric height. In actual
calculations, it is customary to first approximate the trim by using the G m
given by the original waterplane ; and, for a final result, to employ a mean
G m between those corresponding to the approximate and the original water-
planes. Taking an actual case, let it be required to find the draught and
trim in the vessel of Example 2, after discharging 1000 tons of coal and
cargo and loading 300 tons of water ballast. The reduction in displace-

206

SHIP CONSTRUCTION AND CALCULATIONS.

ment is 700 tons, and, assuming the weights to have their centres in the
vertical plane containing b, the centre of buoyancy of the layer —

700
Thickness of parallel layer = =21 inches.

b is found to be one foot forward of the centre of gravity of the original
waterplane, and the leverages of the weights are measured from b. The
work of calculating the trimming moment may be tabulated as follows : — •

Item.

Distance from b.

Forward
Moment.

Aft
Moment.

u c J

n 1

400 tons bunker coal

350 tons cargo from main hold

250 tons cargo aft hold

200 tons W.B. in aft main tank

1 00 tons W. B. in aft tank

14 ft. forward
46 ft. forward

121 feet aft
7 1 feet aft

1 3 1 feet aft

foot tons

3° 2 S°

foot tons

5600

l6lOO

142OO
1 3 IOO

30250 49000
30250

18750

There is thus an aft-trimming moment in operation of 18,750 foot tons.

The moment to alter trim 1 inch in the initial condition is 700 foot
tons } as a first approximation, we therefore get —

Change of trim =

18750

700

27 inches, nearly.

The approximate draughts are —

Forward, 20' o" - (1' 9") - (1' if) = 17' ij" ;

Aft, 26' o"-(i' 9 ") + (i' ii*) = 25' 4"-

Allowing for the difference in the displacement and the metacentric height,
the moment to alter trim 1 inch in the above approximate condition is
found to be 650 foot tons; so that a mean moment is 675 foot tons. Em-
ploying this figure we get with more exactness —

Change of trim ==

i87_ 5 _o
^75

28 inches, nearly.

The final draughts will therefore be-

Forward,
Aft,

17 1 -J - I

17 1 :

2 5 4* + i = 25' 5"

MR. LONG'S METHOD.*— A method of dealing with questions of trim,
which differs somewhat from the preceding one, and which has several

* While Mr. Long makes no claim to the invention of this elegant method of solving trim
problems, his paper contains what appears to be the first published description of it. As seems
fitting, therefore, we have called the method by his name.

MR. LONG'S METHOD.

207

points of advantage, is described by Mr. Long in a paper read recently
before the North-East Coast Institution of Engineers and Shipbuilders.

In this system, use is made of what are called trim lines or curves to
find the trim corresponding to any mean draught and longitudinal position of
the centre of gravity. A trim line is obtained as follows: — First, a level line
is drawn, as W L in fig. 195, to represent the mean draught for which the
trim line is required. On this a point B is taken as the position of the
longitudinal centre of buoyancy at level keel, and a datum line drawn show-
ing its relation, say, to amidships. The horizontal distance from B of the
centre of buoyancy, with the vessel trimming 2 feet by the stern, is then
calculated and marked off at B 2 ; also the distance abaft B of the centre of
buoyancy, with the vessel trimming 4 feet aft, is plotted as at Z? 4 . At B 2 and
£ 4 verticals are erected, and the corresponding trims marked off, the same
scale being used throughout. Through the points thus found and the point
B a line is drawn ; this is the trim line required.

Fig. 195.

Obviously, we have here all the information necessary to determine any
trim up to 4 feet, due to the movement of weights on board ; for, if the
distance the centre of gravity travels aft on account of the movement of
the weights be ascertained and plotted from B along the level line to G,
say, and a vertical be raised to intercept the trim line at D y GO must be
the trim by the stern, as the centre of buoyancy and centre of gravity are
again in the same vertical line.

For forward trims the trim line should be continued below its level line
to indicate the movement of the centre of buoyancy in that direction. It
should be noted that the centre of gravity and centre of buoyancy are here
assumed to travel the same distance when a change of trim takes place.
This is not quite true, as a glance at fig. 191 will show; B being below
G } and therefore more remote from M, moves a greater distance. For quite
accurate work, therefore, the distance plotted from B towards W (fig. 19c)
should be the calculated travel of the centre of gravity plus B G tan

203

SHIP CONSTRUCTION AND CALCULATIONS.

(see fig. 191). It is not necessary to proceed to this refinement in ordinary
cases, as the error thus involved is inappreciable.

One advantage of the trim line system is the absence of formulae. No
calculations are required except a simple one for the travel of the centre of
gravity consequent on the movement of the weights. This will be seen by
an example. It will be interesting to work out Example 1 (page 203) by this
method ; we are able to do this, as fig. 195 is the trim line at the
load draught of the vessel referred to. Employing the figures given, we
get—

Fig. 196.

Travel of centre ot gravity "i 80x150

on removal of weight / " 15814-150 = ?

Plotting this distance from B to F, and erecting a perpendicular to meet
the trim line at G, we obtain FG, or 7I inches, as the trim by stern
required. This is the same result as before.

The trimming weight in the foregoing example is small in comparison
with the displacement, and for such cases we know the ordinary metacentric
method is as accurate as any. Where the weights and moments are great,
however, only approximate results are obtainable by the ordinary method
due to the fluctuating nature of the metacentric height. In this respect the

BILGING. 209

trim line method excels the other, as it is practically accurate for all changes
of trim and draught, however large.

It is perhaps scarcely necessary to point out that a trim line is only
reliable at its own draught, and that where the change of displacement is
considerable, a new curve is required. Experience goes to show that in
ordinary cases the tendency of trim lines is to become less steep with
reduction in draught. For this reason they should be drawn for use-
ful draughts, as those of the load, ballast, and light conditions, and this
would probably be sufficient in most cases. By constructing cross curves
of trim, however, as is done for stability, a trim line for any draught
within the limits of the cross curves can at once be obtained. Such a
diagram obviously provides full trim information for a vessel. Fig. 196
represents the case of the steamer of Example 1. The horizontal lines
are the waterplanes for which trim lines have been drawn. The points in
which the latter intersect their corresponding level waterplanes, and where they
indicate trims of 2 feet, 4 feet, and 8 feet by the stern, and 2 feet by the
head, are enclosed by small circles. Curves through these points give the cross
curves required. If, now, a trim line at an intermediate draught, say of 26
feet, be desired, it is only necessary to draw a level line at this point,
and at heights of 2 feet, 4 feet, etc., parallel lines to cut the correspond-
ing cross curves at A, 5, and 0, a line through these points being
the trim line required.

BILGING. — Given a diagram like that just described, any trim question
relating to the ship for which the diagram is drawn can be readily and quickly
dealt with. Consider, for instance, the important trim problem of finding the
floating condition of a vessel consequent on one or more of her compartments
being bilged and in free communication with the sea. Such a case is depicted
in fig. 197, in which a vessel is shown bilged in the after compartment of the
hold. W\L\ is the line of flotation after the accident, with the ship once
more at rest ; W L the original waterline. The problem is to determine
the line W X L V

Now the change of trim is here caused, not by an added or deducted
weight, but by a loss of buoyancy, and it is usual to treat the problem as
one of loss of buoyancy. By an exercise of imagination, however, the
question may be more easily dealt with ; for, if the hole into the bilged
compartment be assumed closed — the vessel being once more at rest — the
trim will not be affected, but an important change will have taken place
in her floating condition, as the lost buoyancy will have been restored and
the water in the compartment become, for practical purposes, a weight
carried. Viewed thus, it is only necessary to obtain the weight and the
position of the centre of gravity of this water for a complete solution of
the problem. The process of calculation is tentative in character and may
be as follows : — First, the weight of water in the compartment up to the
original waterplane W L should be found, and the parallel sinkage determined
assuming compartment open to the sea and the admitted water placed with

2T0

SHIP CONSTRUCTION AND CALCULATIONS.

its centre of gravity in the vertical plane containing the centre of gravity of
the added layer of displacement. This distance, measured in the trim dia-
gram above the height of the original waterplane, will give the point from
which the level' line and corresponding trim line should be drawn. The trim
can then be obtained, as already described, by finding the travel aft of the
centre of gravity, assuming the weight to be translated to its true position.
This is, of course, a first approximation. It will next be necessary to
calculate the weight of water in the compartment, assuming the surface to
rise to the level of the new draughts, and to use it in the same way in
another trim estimate. If the second approximation should differ much

w--

Fig 197.

from the first, it may be necessary to proceed to a third. But the experi-
ence of the calculator must guide him here.

As a numerical example, take a box-shaped vessel, 210 feet long, 30
feet broad, and 20 feet deep, drawing 10 feet forward and aft; and suppose
an empty watertight compartment at the extreme after-end, 10 feet long, to
be in free communication with the sea. It will be necessary first to draw
out the trim diagram. This is a simple matter owing to the regular nature
of the vessel's shape. We begin by obtaining the trim line at 10 feet

Fig. 198.

draught. A B D, fig. 198, shows the vessel in side elevation, W L is the
level keel water-line, W^U and W 4 L 4 those when 2 feet and 4 feet by the
stern, respectively. Now, assuming the vessel to be floating in salt water,
her displacement is-

210 x 30 x 10

^55

= 1800 tons.

and in passing from the water-line W L to water-line W 2 L% the wedge of dis-

BILGING.

211

placement LSL 2 moves to the position W S W 2 - As S L is half the vessel's
length, and L L 2 i foot, the volume of the wedge is —

105 x 1 x 30 , . -.
= 1575 cubic feet,

and in moving aft, its centre of gravity travels' a horizontal distance g l g 2l

210 x 2

or,

= 140 feet.

The corresponding movement of the vessel's centre of buoyancy is from
B to Z?2> and from what we know of moments, obviously —

1575 x 140

BB 2

1800 x 35

= 3-5 feet.

fig. 199.

The horizontal travel of the centre of buoyancy, with the vessel 4 feet by the
stern, is clearly just double this amount, or 7 feet. This is all the infor-
mation needed to construct the trim line at the initial draught. The trim lines
corresponding to other displacements would be obtained in the same manner.
Fig. 199 is the complete diagram for this vessel, and shows cross curves
with a range from 7 feet 6 inches to 15 feet draught. We are now in a
position to deal with our bilging question.

Beginning with the initial condition, we have —

and,

„. . , r • 1 -, 1 10 x 10 x -jo

Weight of water in bilged compartment = — = 85*71 tons,

2T2 SHIP CONSTRUCTION AND CALCULATIONS.

Parallel sinkage assuming water situated ^ ^

amidships and compartment open to,- = — = 6 inches,

1 I JO ° x 3°

the sea ;

also,

Horizontal travel* aft of vessel's centre of j

gravity, assuming the water at the in- 1 90 x 100 _ , ~
creased draught to move into its true ~~ 1890
position and the ship's bottom to be intact

Referring to fig. 199, we can draw at once the trim line corresponding to
a level line at 10 feet 6 inches, and by measuring 4*76 feet along this level
line from the vertical AB, and erecting a perpendicular, we get 2 feet 10 J inches
as the trim by the stern. The draughts of the vessel will thus be —

forward, 100+0— 15^=9 o£ ;
Aft, 10' o" + 6" + 1' si" = 11' "F-

In the second approximation, we start with the vessel in this trim. The
weight of water in the bilged compartment will now be —

1 1 "86 x 10 x ^o

— — 101*66 tons.

35
The

„ „ n . , ior66 x 2C x 12 ,o., ,

Parallel sinkage = — = 64 inches nearly,

210 x 30

and taking the centre of gravity of the water at the middle of the length
of the compartment, Ave get as before —

Travel of vessel's centre of gravity duel ioi"66 x 100

, . . c = — = S'^S feet aft.

to admission 01 water J iroT'66 "

From the trim diagram we find the corresponding trim by the stern to be
3 feet 2 J inches.

Dividing this equally forward and aft, and adding 6| inches as the
parallel sinkage, the draughts become —

Forward, 10' o'' + 6 J - (1' 7^") = 8' n§";
Aft, 10' o" + 6 1 + (i' 7f") = 12' 2 J".

By a third approximation we obtain the draughts —
Forward, 8' n"
Aft, 12' 2f,

in which condition the vessel will float in equilibrium whether the after compart-
ment be now open to the sea or not. Of course, the same result could be
obtained by the ordinary method, i.e., by calculating the height of the longi-
tudinal metacentre, the moment to alter trim, and the heeling moment due to

* The trimming of the vessel causes die water in the compartment to change level, and a
small quantity of the water to move aft ; this affects the position of the ship's centre of gravity,
and therefore the trim, but to no appreciable extent, except in the case of large compartments.

APPROXIMATE CALCULATIONS. 2* I $

the admission of the water, and finally dividing the latter by the moment

to alter trim. We do not propose to deal with the problem in this way,

as the principles involved have already been fully explained. The student,
however, should work it out himself as an exercise.

APPROXIMATE CALCULATIONS.— Although a commanding officer may
know nothing regarding his vessel beyond her dimensions and displacement,
he is still able to estimate, roughly, at least, the trimming effect due to the
addition, removal, or movement of weights. In the formula —

• , W x Gm c

Moment to alter trim i inch = , loot tons,

12 x L

if we assume G m to be equal to L, which is roughly true in the case ot
ordinary cargo vessels at their load displacements, the trimming moment per

inch becomes foot tons.

I 2

Applying this simple formula to the example on page 198, we get —

and

2000
Moment to alter trim 1 inch = - — - — 166*6 foot tons,

So x 80 .

Change of trim = " rr , = 24 inches,

which compares with 25 inches obtained by the exact method.
In the case of Example 1, page 203,

15814

Moment to alter trim 1 inch — = 1318 foot tons.

12 °

For the effect of discharging 150 tons from a position 80 feet before
the centre of gravity of the load waterplane, we thus have —

150 x 80 .

Trim by stern = — q ~ = 9 inches, nearly.

The ordinary formula gives 7J inches ; the approximation is thus near
enough for practical purposes, an inch or two either way, in ordinary cases,
not being of great importance. As the value of Bm rises rapidly with
reduction in draught, the formula is inapplicable for draughts other than the
load draught. Also, it is unsuitable in the case of vessels which are of
abnormally shallow draught in relation to length, as G m and L are then far
from being even approximately equal.

An approximate formula, giving closer results than the foregoing, has
been devised by M. Normand.* By this rule, for the height of the longi-
tudinal metacentre above the centre of buoyancy in ordinary cargo steamers,

we have —

A- x L
Bm = -0735 ~ b ~xT feet '

* See a paper in the Transactions of the Institution of Naval Architects for 18S2.

214 Srilt> CONSTRUCTION AND CALCULATIONS.

where A = area of load waterplane in square feet.

L = length on the load waterline in feet.

b = breadth of ship amidships in feet.

V = volume of displacement in cubic feet.
Assuming Bm = G m,

V A 1 x L

Moment to alter trim i inch = — - x '0735 , —~n

L x 12

A 2

= -000175 t- foot tons.

This is Normand's formula.

Now, if T be the tons per inch of immersion,

420
A = 420 T
A 2 = 176400 P.
Substituting this value of A 2 , the formula takes the convenient shape —

30 'o x 7~ 2

Moment to alter trim 1 inch = r foot tons.

Applying the Rule to Example 1, page 203, we get —

- , 30*9 x SV3 x SVS
Moment to alter trim 1 inch = ^— ^ ^-f ^^

= 1567 foot tons;
the previous value being 1570 foot tons.
In the case of Example 2 —

Moment to alter trim 1 inch = ^— y — = 701 foot tons,

which compares with 700 tons, the exact value.

Besides the foregoing, we have in the trim-line method a ready means
of making approximate estimates of trim. It happens that the trim lines in
ordinary cases are practically straight, and make certain definite angles with the
corresponding level lines, also that these angles, in different vessels of similar
type, are about the same at corresponding draughts. It has been suggested,
therefore, that in type vessels a note should be made of the trim angles at
useful draughts, such as those of the load, ballast, and light conditions.
The trim line in the case of any new design could then be plotted at once
with sufficient accuracy for preliminary calculations. If only one trim should
be required and the angle of the corresponding trim line is known, it can be
found quickly by means of the formula —

Change of trim in inches = — ^ — x C x 12,

where W = weight shifted,

d — distance shifted,
W = whole displacement,
Q = tangent of the trim line angle.

APPROXIMATE CALCULATIONS. 2t$

C varies considerably with type of vessel. Ships of very light draught relatively
to their length, have flatter trim lines than those of ordinary proportions, but
an average value at the load draught of cargo steamers 300 to 500 feet in
length, and of the usual fullness, is "9163, corresponding to a trim line angle
of 42 J°. Assuming this trim line angle in the case of Example 1, page 203 ;

_. . 150 x 80

Change of trim = — 5 x •016-? x 12 =8 '4 inches,

to 15814- 150 J ° t i

which is a good approximation. The student should apply the rule in
other cases ; an officer might try it on his own ship.

TRIM INFORMATION FOR COMMANDING OFFICERS.— An important
use to which the trim line method may be applied, is the supplying of in-
formation to masters and others who have to deal with loading and ballast-
ing operations. With a good-sized diagram, showing the trim curves of his
vessel, and a scale, a master should be able to decide in a few minutes
any question of trim, provided the weights to be shipped and unshipped,
and their movements, be known. We have already fully explained the pro-
cedure. If builders would supply such trim diagrams to new vessels, with
instructions as to their use, we are confident they would come to be
highly appreciated.

QUESTIONS ON CHAPTER VIII.

1. Define the longitudinal metacentre. A prism of rectangular section 200 feet long,
and 33 feet broad, floats at a draught of 11 feet forward and aft, calculate the height of
the longitudinal metacentre above the centre of buoyancy.

Ans. — 303 feet.

2. The equidistant ordinates of a vessel's waterplane measured on one side of the
middle line are — "2, 7"2, I0'6, I2"0, 12*0, I2'0, 10*9, 9/6, and 1 9 feet, and half-oi'dinates
at the ends have values 3*8 and 77 feet, respectively; find the height of the longitudinal
metacentre above the centre of buoyancy, the volume of displacement being 20,000 cubic
feet, and longitudinal interval between the ordinates 15 feet.

Ans. — 102-26 feet.

3. Obtain the expression giving the change of trim consequent on moving a small
weight w tons longitudinally through a distance a feet. A vessel 300 feet in length
floats at a level draught of 17 feet; she has a longitudinal metacentric height of 400 feet,
and a displacement of 4500 tons ; a weight of 50 tons is moved aft through 100 feet ;
find the new draught forward and aft.

f Forward, 16 feet, 7 inches.
Ans. -Draught ^ J? ^ $ ^^

4. Deduce the moment to change trim one inch in the case of the vessel of the
last example.

Given that the tons per inch of immersion is 30, calculate the new draught forward
and aft when the following weights have been placed on board in the positions named.

216 SHI** CONSTRUCTION AND CALCULaI'IONS.

Weights Tons). Distance from Centre of Gravity of Waterplane.

20 lool

45 80 -before.

15 40)

60 50]

40 80 J- abaft.

30 1 10 J

Ans. — Moment to change trim one inch, 500 foot tons.
(Forward, 17 feet, 3f inches.
Draught | Aftj I7 feetj lQi inches<

5. Suppose a weight of moderate amount to be put on board a vessel, where must
it be placed so that the ship shall be bodily deeper in the water without change of trim?

Describe, clearly, why it is that vessels in passing from salt water to fresh water
usually change trim slightly as well as change their draught of water.

6. It is desired that the draught of water aft in a steamship shall be constant,
whether the coals are in or out of the ship. Show how the approximate position of the
centre of gravity of the coals may be found, in order that the desired condition may be
fulfilled.

7. What is a trim line? Describe how such a line is obtained, and explain its uses.

8. A box-shaped vessel, 260 feet long, 40 feet broad, and 25 feet deep, floats at an

even draught of 20 feet, construct the trim line for this draught. If 100 tons be shipped

aft, with its centre 100 feet from amidships, find the new draught forward and aft, using

the trim line.

f Forward, 19 feet, 6f inches.
Ans. -Draught | Aft _ 2[ ^ ^ ^ches.

9. Referring to the vessel of the previous question — if a watertight compartment

situated at the extreme after-end be bilged and in free communication with the sea, what

will be the new floating condition, the bilged compartment being 10 feet long, the full

breadth of the vessel, and with half its space occupied by cargo?

(Forward, 19 feet, 2| inches.
Ans. — Draught \ . - r , .. . .

fa I^Aft, 21 feet, 7i inches.

10. It is desired that a certain vessel shall float with any two compartments in open
communication with the sea. Describe in detail the calculations involved.

11. A steamer 330 feet long, 48 feet broad, drawing 24 feet aft, and 20 feet forward, has
to cross a bar over which there is a depth of water of 23 feet, 6 inches. The vessel has a
fore-peak tank with a capacity of 100 tons. Given that the centre of gravity of this tank
space is 150 feet forward of the centre of gravity of the load-water plane, find, by an approximate
method, if filling the tank will modify the draught sufficiently to admit of the vessel cross-
ing the bar. The displacement in tons to begin with is 78CK).

12. Referring to the previous question, if it be given that the longitudinal metacentric
height is 345 feet, and the tons per inch of immersion 33, estimate correctly the effect on the
draught of filling the fore-peak tank.

/'Forward, 21 feet, 2 inches.
Ans. — Draught -. , . . . ,

& l^Aft, 23 feet, 4 inches.

CHAPTER IX.

Stability of Ships at Large Angles of Inclination.

IN Chapter VII. we saw how to obtain the righting or upsetting moment
for a vessel when inclined through initial angles about the upright
position. We learned that up to angles of 10 or n degrees, the meta-
centre may be considered as fixed in position, and that inside these limits the

Heeling Moment * W x G M x Sin. Q.

Thus, taking the two cargo vessels for which the values of G M were ob-
tained, we have —

Righting moment at 10 degrees (small vessel) = 4525 x 1-73 x '1736

= 1359 foot tons.
Do. do. (large vessel) = 15814 x "85 x '1736

— 2 333 f° ot tons.

Fig. 200.

For inclinations much exceeding 10 to 12 degrees, however, except in
the instance of a single type of vessel, the righting moment cannot be ob-
tained by this method. The exception referred to, is where a vessel is so
designed that all the immersed cross sections are circular segments with a
common centre in the middle line. We have already shown that for floating
bodies of this form the line of upward pressure passes through the same
point M for all transverse inclinations, so that, if there is no disturbance in
the weights, the distance GM will remain unchanged as the vessel heels from
angle to angle. Fig. 200 represents a vessel of constant circular section in-
clined to some angle Q t G M is the metacentric height, and if W be the
displacement, we have, by applying the metacentric method—

217

!l8

SHIP CONSTRUCTION AND CALCULATIONS.

Righting moment in foot tons = W x 6 M x Sin.

= W xGZ,
G Z being the arm of the righting couple.

The only variable in this expression is the sine of the angle ; therefore,
to construct a stability curve for this simple case, it is only necessary to
draw a horizontal line, set off on it, to a convenient scale, the various angles
at z 5j 3°> 45) etc., degrees, erect perpendiculars at the points of division, on
these perpendiculars scale off the various values of righting moment as ob-
tained above, and draw a fair curve through the points so found.

As a specific case take a cylindrical vessel, 20 feet in diameter, floating
with its axis in the waterplane. We shall deal only with the levers or right-
ing arms, so that the length of the vessel is immaterial. If we assume
the centre of gravity to be 2 feet below the centre of the figure, we may,
using a table of sines, at once write down the value of the righting arm
for any inclination. Obviously, the righting arm increases from zero at o
degrees to a maximum value at 90 degrees, and thence gradually decreases

Fig. 201.

♦5 ' «o n ♦« w3 t£5

GIQPK.«J OF INCLINATION

to zero again at 1S0 degrees ; obviously, too, by drawing a diagram, the
righting arm or lever at an angle Q, say, is the same as at the angle
180 - Q. It is, therefore, only necessary to calculate values from o degree
to 90 degrees; and at intervals of 15 degrees, which is close enough for the
purpose of constructing a curve, these are as follows : —

Inclination,
in Degrees.

J 5

3°
45
60

75
90

Sine of Angie.

•258S

•5
7071

•S66
"9 6 59

Righting" Arms,
in Feet.

'5176

I'OO

1-414

1732

1-932

2 'OO

Fig. 201 is the curve constructed from this data. On a little con-
sideration it will be seen to represent, for every draught, the curve of
righting arms of all vessels of circular section, whatever their length or
diameter, in which G M = 2 feet ; and since the righting moment at any
angle is equal to G Z, or the ordinate of the curve at that angle, multiplied

STABILITY OF SUBMARINES.

219

by the displacement, if the scales be always altered to suit, this curve will
also represent the righting moments of all vessels of all circular section having
a metacentric height of 2 feet

SUBMARINE VESSELS.— The cylindrical vessel just referred to is
assumed to float with part of its bulk above the surface — the case of

Fig. 202.

WATER SURFACE.

Fig. 203.

WATER SURFACE.

ordinary vessels ; but when properly designed, a vessel may have stability
even when totally submerged. The submarines, now so much in evidence,
are examples in point. A stability curve of a submerged vessel may be
easily obtained by a method analogous to that employed in the previous
case. Figs. 202 and 203 show a submarine floating upright and heeled,
respectively. B, the centre of buoyancy, is also the centre of bulk ; G is
the centre of gravity. Here, G being below B, when the vessel is heeled
as in fig. 203, the tendency of the resultant forces of weight and buoyancy

220

SHIP CONSTRUCTION AND CALCULATIONS.

is to restore the vessel to the position in fig. 202. If G were above B,
and the vessel then inclined, the tendency would be to heel still further
until G became vertically below B — the position of stable equilibrium.

Applying the formula BM=-.j, since / = o, BM is zero, and therefore,

B and M are coincident. Thus, in this special case, B and B G have
much the same functions as M and GM in the preceding one, for at any
angle 6 the righting or upsetting moment = B G Sin ; so that, as in the
case of the cylinder floating at the surface, the lever varies directly as the
sines of the angles of inclination, has zero values at o degrees and 180
degrees, and a maximum value at 90 degrees. Clearly, if BG is 2 feet,
fig. 20 1, the stability curve for a cylindrical vessel floating on the surface
may also be taken to represent the curve of righting arms for the sub-
merged vessel.

Fig. 204.

Fig. 205.

A noteworthy point in curves of righting arms of totally submerged
vessels is that they represent the stability when inclined in any direction,
either transverse or longitudinal, which follows from the fact that the line
of buoyancy must always pass through the same point, viz., the centre of bulk.
This is, of course, by no means the case in vessels floating at the surface,
which have enormous righting power when inclined longitudinally.

VESSELS OF NORMAL FORMS.— The case of an ordinary vessel, it
need hardly be said, admits of no such simple treatment as those just
dealt with, owing to the increased difficulty of obtaining values of G Z.
Fig. 204 shows, in cross section, an ordinary vessel floating upright at a
waterplane W L. M is above G, therefore the condition is one of stable
equilibrium. Fig. 205 shows the same vessel heeled to a large inclination.
The movement has caused the centre of buoyancy B to travel out to B h
through which the resultant buoyant pressure now passes. No weights are
supposed to have been shifted during the heeling, so that the centre of
gravity G is unchanged in position. Unlike the case of the cylinder, the
line of upward force does not intersect the middle line at the metacenire,

VOLUMES AND MOMENTS OF WEDGES. 221

G M in fig. 204 and G A in fig. 205 having different values. Obviously,
then, the equation —

Righting Moment = W x GM x Sin Q
is no longer applicable, and in order to obtain the values of righting arms
or righting moments at large inclinations we must resort to another method.
In this case, to find the lever GZ between the verticals through G and
B when the vessel is inclined to any angle, we must proceed as follows : —
Referring to fig. 205, as previously pointed out, the transference of the
wedge of displacement from WSW V to LSL it compels the centre of buoy-
ancy to travel from B to B x . A line joining these points is parallel to the
line joining $$<& the centres of gravity of the wedges, and

^ Volume of wedge x g 1 g 2 ^
1 Volume of displacement*

Now, through B draw a horizontal line to cut the verticals through G and
B 1% in N and/?; and from g x and g. 2 drop perpendiculars g Y h b g 2 h 2 on W\ L x ;
then clearly,

Volume of wedge x h x /z 2
"~ Volume of displacement"

Also, BR - BN = NR = GZ;

and BN = BG sin 0;

r. -, Volume of wedge x h y h.-> n ~ .

so that, GZ - ^r^ , ,. ° -f - BG sin. 0;

Volume 01 displacement

this is the value of the stability lever required, and the equation is known
as Atwood's formula. The only portion of this expression which cannot be
quite easily obtained, is U x h\h^ the horizontal moment due to the transverse
movement of the wedge of displacement, and we now propose to show how
this is calculated, and the stability lever or moment arrived at.

VOLUMES AND MOMENTS OF WEDGES.— A body plan of the ship
is prepared with transverse sections, spaced as for a displacement calculation,
and with radial planes drawn to represent the floating condition of the vessel
when upright, and when inclined at all the inclinations required to give data
for the construction of the stability curve. The sections should represent
the full volume available for buoyancy, and be drawn to the top watertight
deck and to the outside of the shell-plating. With regard to the radial
planes, it is found convenient to draw them so as to intersect in the middle-
line plane {see 0, fig. 206), although this does not usually ensure that the
in and out wedges shall be of equal volumes, as, of course, they must be;
but it allows all the inclined planes up to any maximum inclination to be
drawn at once, while the correction due to the inequality of the wedges can
easily be made afterwards.

To furnish spots close enough to obtain the correct form of the stability
curve, the radial planes should be drawn at intervals of about i o to 15
degrees. Discontinuities in the vessel should be carefully dealt with. The

222

SHIP CONSTRUCTION AND CALCULATIONS.

entrance into the water of the deck edge, for instance, causes a sudden
change in the form of the immersed wedge, and to ensure accuracy in find-
ing the volumes and moments of wedges by Simpson's Rules, a radial plane
should occur at this point, with a suitable number of radial planes on each
side of it.

These particulars attended to, the measurement and tabulation of the
ordinates, or breadths of sections at the various radial planes on each side
of the point 0, is proceeded with, those of the immersed side being kept
separate from the emerged, and those of the various planes separate from

Fig. 206.

each other. At each radial plane, the volume of an elementary wedge and
its moment about a fore-and-aft horizontal axis through is next calculated.
To show how this is done, let b be the length in feet of an ordinate
of a radial plane, say, on the immersed side; then the sectional area at this
ordinate of a very small wedge of the immersed volume, treating it as a

b 2

segment of a circle, will be - 9 square feet, Q being the circular measure

of the wedge angle; and if t be the thickness in feet of a thin transverse

a
slice, its volume in cubic feet will be - b 2 1

VOLUMES AND MOMENTS OF WEDGES.

223

Having obtained such values for slices at various sections in the length
of the wedge, to find the volume of the latter becomes simply a matter of
finding the area of a plane surface, for, if a base line representing to scale
the length of the vessel be taken, and at points corresponding to the

positions of the various sections the quantities - b 2 t be set off as rect-
angles, each on the little quantity t as base, and a curve be drawn through
the tops of the rectangles, an area will be enclosed representing the sum of
the volumes of all the slices into which the wedge may be supposed divided,
and, therefore, the volume of the whole elementary wedge.

The moment of an elementary wedge may be similarly dealt with. For
instance, taking the same radial plane and ordinate, the distance from 0, of

Fig. 207.

^^"^-T n

5 S

S 6

S 9

DEQftEE.3 0FWUNAT10N

the centre of gravity of a thin transverse slice of the elementary wedge is

- b feet, and the geometrical moment of the slice in foot units about a fore-

and-aft axis through 0,

-6 x -b 2 t, or ^b s t
3 2 3

To express the whole moment as an area, it is only necessary to plot, at
the same points in the length as in the case of the volume, rectangles, each

on a base £, giving the various values of - 6 s £, and draw a curve. The

volumes and moments of the elementary wedges may now be found by cal-
culating the above areas.

It is next necessary to combine the figures of the elementary wedges
to obtain those of the full wedges of immersion and emersion. We shall
show presently how this is done in an actual case, but it may also be
explained graphically. Take a base line A B (fig. 207), and let it represent

224

SHIP CONSTRUCTION AND CALCULATIONS.

on some scale the circular measure of the maximum wedge angle. Mark
off points at the various angles at which the volumes of the elementary
wedges have been calculated, and plot rectangles, each on a base 9 (9 =
the circular measure of the elementary wedge angle), representing the volumes
of the corresponding elementary wedges. A curve through the tops of these
rectangles will enclose an area A C D B, representing the sum of the volumes
of all the elementary wedges, that is to say, the volume of the whole wedge.
To find the volume of any wedge within the limits of A C D B, it is
simply necessary to plot an ordinate at the correct wedge angle, and by
Simpson's Rules calculate the area thus cut off. Separate diagrams are
necessary for the immersed and emerged wedges. Coming to the moments
of the full wedges, it must be noted that while the moments of the
elementary wedges are, in the first instance, calculated about a longitudinal

Fig. 208.

15
DECREES OF INCLINATION

axis through (the point of intersection of the radial planes) the moments
required for statical stability are taken about a longitudinal vertical plane
through (see y y, fig. 209), and, therefore, in combining the elementary
moments to obtain those of the full wedges of immersion and emersion,
each of the former has to be multiplied by the cosine of the angle which
the particular elementary wedge makes with the horizontal. For instance, in
calculating the moment of wedges of 30 degrees, the moment of the elementary
wedges at o degrees about a longitudinal axis through has to be multiplied
by the cosine of 30 degrees, and those at, say, 15 and 30 degrees, by the
cosine of 15 and o degrees, respectively. Fig. 208 is the complete diagram
of moments, the abcissse being in circular measure, A B representing 30 de-
grees. Since the sum of the moments is required, the diagram takes
account of the wedges on both sides of the axis y y. Thus the little rect-
angle at AC is the sum of the moments of the in and out elementary
wedges at o degrees multiplied by cosine 30 degrees; the rectangle at EF

VOLUMES AND MOMENTS OF WEDGES.

225

the sum of the moments of the in and out elementary wedges at 15 degrees
multiplied by cosine 15 degrees; and the rectangle at DB the corresponding
quantity at 30 degrees multiplied by cosine o degrees. Clearly, from our
preceding remarks, the whole area A G D B represents the sum of the moments
of the immersed and emerged wedges at 30 degrees about the axis y y.

It will be seen that, in the case of the moments, a new diagram is
required for each inclination at which the righting arm or moment is
calculated, as the elementary wedge at the limiting inclination must always
be multiplied by cosine o degrees, and the others by the cosine of the
angle which each of them makes with the limiting radial plane.

Such are the principles to be followed in finding the volumes and
moments of the various in and out wedges, and they are seen to present

Fig. 209.

no greater difficulty than is involved in the application of Simpson's Rules
to the calculation of plane areas.

CORRECTION OF WEDGES.— It must not be forgotten that the moments
of the various wedges, found as above, have to be corrected on account of
the immersed and emerged wedges, as drawn in the body plan, being un-
equal in volume. This may be done as follows : — Suppose the immersed
wedge is in excess, then the vessel is shown deeper in the water than she
should be, and the vertical distance between the true and the assumed water-
planes, or —

Thickness of layer = -7 ■?—. — ~. — — = —

J Area of inclined waterplane

where V x and V 2 are the volumes of the in and out wedges, as drawn.
P

226

SHIP CONSTRUCTION AND CALCULATIONS.

Let fig. 209 represent the case dealt with; let WL be the upright water-
plane, W\Lx> the uncorrected inclined plane, and W 2 L 2 the corrected inclined
plane. Since the correct wedges are W S W 2 and L 2 SL, the moment of the
volume W\ S W 2 is to be added, and that of L x S L 2 deducted, from the
moments of wedges as calculated. Call volume W x 0SW 2 V h and volume
L x S L 2 u 2 , and let od x and od 2 be the distances of their centres of gravity
from axis y y t then the correction is —

u x xod x -v 2 xod 2 (1).

If the centre of gravity of the whole layer W\ L x L 2 W 2 be at a distance x
on the immersed side of the axis —

u l xodi-u 2 xod 2 ^ (Vi + u 2 ) x t

Fig. 210.

and equation (1) will be negative, and the correction a deduction. If x
be on the emerged side, (1) will be positive, and the correction an addition.
Now, suppose the emerged wedge to be in excess (see fig. 210). In
this case, the moment of the volume W x S W 2 will be deducted from, and
that of volume L x S L 2 added to, the calculated moment of the wedges.
Using the same symbols, we have — ■

Correction = t/ 2 x od 2 -v x x od { (2)

and this is also equal to (u x + U 2 ) X, where x is the distance of centre
of gravity of the whole layer from the axis y y. If x be on the emerged
side, equation (2) will be negative, and the correction a deduction ; if on
the immersed side, positive, and the correction an addition. A little con-
sideration will make this quite clear. Rules for the correction of the
moments of wedges may now be stated as follows : —

1. If the immersed wedge be in excess, and the centre of gravity of

CORRECTION OF WEDGES. 227

the layer on the immersed side of the axis of moments, the cor-
rection will be a deduction; but if it be on the emerged side, an
addition.

2. If the emerged wedge be in excess, and the centre of gravity of
the layer on the emerged side, the correction will be a deduc-
tion, but if it be on the immersed side, an addition.

In most cases the layer is small, and the centre of gravity of the in-
clined plane may be used for that of the layer. This simplifies the work,
but if the layer be large, its centre of gravity must be calculated, and its
correct distance from the axis employed. Thus we arrive at the value of
the quantity u x h\ h 2 at any inclination, and by Atwood's formula the length
of the stability lever may be written down.

As a practical example, let us obtain the stability curve for an actual
vessel, such, for instance, as the large cargo steamer whose dimensions and
other particulars are given on page 182.

This vessel, laden to her full draught, has, with a certain distribution
of cargo, a metacentric height of '85 feet. The centre of gravity is 22*65
feet above the base line, and the centre of buoyancy 14*4 feet above the
same line, so that B G, the distance between the centre of buoyancy and
centre of gravity, is 22*65 -14*4, or ^' 2 5 f eet - Let this be the basis of
our calculation.

Fig. 206 shows the body plan drawn out as already directed, with trans-
verse sections and radial planes, the former showing the vessel's shape at
each tenth part of the length, and also at intermediate positions towards the
ends, and the latter being drawn at intervals of 14J degrees so as to ensure
a radial plane striking the deck edge, which becomes immersed at 29 degrees.

Before starting to measure the ordinates, sheets must be prepared (see
Table I.) with a suitable number of columns to take the calculations for
the areas of the radial planes and the functions for the volumes and moments
of the elementary wedges. Two such sheets are required for each radial
plane, the immersed and emerged sides, as previously mentioned, being kept
separate. In Table I. we give the calculations for the elementary wedges
on each side of the axis at 29 degrees inclination. As the work is the
same for each elementary wedge, the method followed is amply illustrated in
this table, which is drawn up in accordance with explanations given for the
general case.

Having obtained the requisite information for the various elementary
wedges, it is utilised to determine the values of the righting arms when
inclined to angles increasing by increments of 14-J- degrees.

Let us deal with the vessel when inclined, say, to 29 degrees. The
elementary wedges required are those at o degrees, 14^- degrees and 29 de-
grees, respectively. The information is combined, as in Table II., which is
seen to consist of the numerical work entailed in deducing the areas of
the volume and moment diagrams previously described.

22>

SHIP CONSTRUCTION AND CALCULATIONS.

TABLE I.

Elementary

Wedge

29 Degrees, Emerged

Side.

Products

Products for

Ords.

£S

lor

Ords. 2

Volume of

Ords. 3

Moments of

Area.

El. "Wedge.

El. Wedge.

—

io'S

2 1'0

220

II58

2316

i9'5

29*2

380

570

7415

III22

29-8

II9'2

888

3552

26463

IO5852

3*'9

6 3 -8

1018

2036

32462

64924

3 1 '9

127*6

1018

4072

32462

I29848

3 r 9

6 3 *8

1018

2036

32462

64924

3*"9

127*6

1018

4072

32462

I29848

3**9

6 3 -8

1018

2036

32462

64924

30-8

123*2

949

379 6

29218

H6872

20*5

3°'7

420

630

8615

I2922

9±

II'2

2 2 - 4

125 |

250

1405

28lO

■

3) 79 2 '

264*1

3) 2327c

7757

3 J706362
2 35454

LEMENTARY

Wedge

29 Degrees, Immersed

Side.

Products

Products for

Ords.

for

Ords. 2

Volume of

Ords. 3

Moments of

Area.

El. Wedge.

—

26*1

52*2

681

I362

17780

3556o

28*8

43' 2

829

1243

23888

35S32

306

122*4

93 6

3744

28652

I I4608

310

62*0

961

1922

29791

59582

3 1 ' 1

124*4

967

3868

30080

I2032O

3 1 ' 1

62'2

967

*934

30080

60160

3 I# i

124*4

967

3868

30080

I2032O

31-1

622

967

r 934

30080

60160

3°'7

122*8

942

3768

28934

H573 6

24*S

1 A

37*2

015

922

!5 2 53

22879

i3'4

26*8

1S0

360

2406

4812

3) 839-9 3) 24925

, J749969

279*9

8308 249989
Emerged wedge 235454

Both wedges 485443

SPECIMEN STABILITY CALCULATION.

229

TABLE II.

Calculation for Stability (Statical) at 29 Degrees Inclination.

Immersed Wedge.

Functions of

Cubes
(both sides).

S.M.
I

4
1

Products of
Functions of

Cubes
(both sides).

Cosines of

Angles of

Inclination.

Functions for
Moments
of Wedge.

Functions
of Squares.

S.M.

Functions

for
Volumes.

6225
6746
8308

4
1

6225

26984

8308

332636
3645 6 4
485443

332636

1458256

485443

•8746
■9681
I 'OOO

290923

1411737

485443

Emerged wedge

J Ang. interval

Long, interval

4*S l 7

39994

2)^523
761
•084

63-92
46-93

2li

SI03

J Ang. interval

729367
•084

61266-8
Longitudinal interval 46*93

Moment (uncorrected) 2875251
Correction (deduct) 1518

Im. wedge in excess 3000 cubic ft. Vol. of displacement 553490)2873733
Stability corr. 3000 x -506 = 151S RN - ■

BG Sin. = 8*25 x -4848 = 4-0

GZ = 1*19 feet

Emerged Wedge.

Ill

Functions

of
Squares.

S.M.

Functions

for
Volumes.

6225
65°3

7757

4
1

6225
26012

7757

Function of area of W.P. at 1 ,™ , . T ,

• /t a a \ f 2 79'9 (Table I).

29 (Immersed side) ) '

Function of area of W.P. at

29 (Emerged side)

| 2 6 4 -i (Table I).

39994

Longitudinal interval
Total Area of W.P.

544'°
46-93

2 553° square ft.

Functions of Ords. 2 of W.P. at 29 (Im. side) 8308 (Table I.)
Functions of Ords. 2 of W.P. at 29° (Em. side) 7757 (Table I.)

2 ) 55*
275-5
Longitudinal interval 46*93

2553° ) I2 9 2 9
C.G. of W.P. towards immersed side '506 feet.

Righting arms estimated in this way for the the three inclinations above
mentioned are given in Table III., and the corresponding curve of stability
in fig. 211. Of course, for the righting moment at any inclination, the

230

SHIP CONSTRUCTION AND CALCULATIONS.

ordinate given by fig. 211 must be multiplied by the displacement in tons,
and this information is also included in Table III.

TABLE III.

Righting Arms and

Righting Moments.

S.S. 469' 4" x 56' 0"

/ ft

X34 10

Displacement, 15^

>i4 tons.

Magnitude

Righting

Righting '

of Wedges

Arms in

Moments in

in Degrees.

Feet.

Foot Tons.

•34

5376

I'20

18977

43i

2'59

40958

2-94

46493

72^-

2-13

33 68 3

•81

12809

Fig. 211

;

'£.

■

I-Q

___— — -""'I

3 11^

Vii

In plotting a stability curve the correct contour near the origin is
readily obtained if the tangent to the curve at that point is drawn. This
may be done as follows: — At a point on the base line indicated by 57*3
degrees, erect a perpendicular and mark off on it, to the same scale as the
righting levers, a distance equal to G M, the metacentric height. Join the
point thus obtained with the origin of the stability curve. This is the
tangent required, and it will be found that the curve will tend to conform
with this line as it nears the origin. As an example, the stability curve of
fig. 211 is reproduced in fig. 212, and the tangent to the curve at the
origin is drawn as described, the metacentric height in this being '8$ feet.
The explanation of the foregoing is as follows : —

TANGENT TO STABILITY CURVE AT ORIGIN.

23I

and

By the metacentric method —

GZ = Gt
GZ Gi

Now the denominators of these fractions are in circular measure ; let them
be expressed in degrees, Q becoming a\ and 5 7 '3 degrees being substituted
for i, there being that number of degrees in the angle whose circular
measure is 1 ; the equation may now be written —

GZ _ GM
«° 57'3°'
Referring to fig. 212, if G Z be the stability lever at a point close to
the origin of the curve, and a° the distance in degrees measured along

Fig. 212.

fr<f*i

573

the base line up to this lever, the small portion Z of the curve will lie
in a straight line, tangent to the curve at this place.

The tangent of the angle which this line,
and therefore the stability curve near
the origin, makes with the base

GZ
a°

57'3°

which is the value employed above in setting off the tangent to the curve
at the origin.

CROSS CURVES. — The foregoing method of obtaining curves of stability
is seen to involve considerable calculation. It has also another drawback.
For most vessels several stability diagrams are required. Such curves should,
indeed, be available for at least the four conditions referred to when deal-
ing with the metacentric height, viz., the launching, light-ship, ballast, and
fully-loaded conditions. By the above method we should thus have four
troublesome calculations. And, moreover, if the vessel happened to be loaded
or ballasted to draughts other than those originally allowed for, it would
not be possible to ascertain her condition as regards stability without fresh
calculations.

232

SHIP CONSTRUCTION AND CALCULATIONS.

This defect in Barnes' method, as that by means of the wedges is called,
was quickly seen when, a good many years ago, scientific attention was turned
in earnest to this important subject. Many ingenious schemes were pro-
pounded for arriving quickly at the knowledge of a vessel's stability under
all conditions of draught and lading, of which the best and simplest, and
most generally employed, is that known as the cross-curve method. Here
the abscissae of the stability diagram is in terms of displacement, instead of

Fig. 213.

Too >

egrees of inclination, as in the ordinary case. In the complete diagram
there is a series of curves, each exhibiting for one inclination variations in
the righting arms or righting moments, as the case may be, with change in
the displacement.

In fig. 213 a cross-curve stability diagram is depicted, with curves at
15, 30, 45, 60 degrees, and so on. The great value of this diagram is that
it supplies us at once with the stability at every displacement or draught,
and every inclination from the upright within the limits of the calculation.
For example, suppose we require to know the vessel's stability when floating
at a draught corresponding to a displacement of, say, 4000 tons ; it is only
necessary to draw a line at this point in the scale of tons perpendicular to
the base of the diagram, to measure the distances A B, AC, AD, etc., cut
off by the curves at 15, 30, 45 degrees, etc., to set off these distances as
ordinates in a diagram having degrees of inclination as abscissae, and draw a
curve through the points so obtained. The result, subject to a correction, is
an ordinary curve of stability such as might be obtained by Barnes' method

CROSS CURVES.

233

(see curve A, fig. 214). We thus see that the cross curves lie in planes
perpendicular to those of the ordinary curves, and it is from this circum-
stance the former derive their name.

The relation between these two sets of curves may be simply illustrated

Fig. 214.

DEGREES OF INCLINATION

as follows :— Take a model representing half a solid cylinder, and assume it
to be cut by planes perpendicular to the base and parallel to the longitu-
dinal axis; these will intersect the curved surface of the model in straight
lines. Next, suppose it to be cut by planes perpendicular to the base and

Fig. 215.

w /

A tL

to the longitudinal axis; the lines of intersection with the model surface
will obviously be half circles. If these half circles are considered to be
ordinary curves of stability, the straight line intersections will be the cor-
responding cross curves.

234

SHIP CONSTRUCTION AND CALCULATIONS.

Fig. 216.

CORRECTION FOR CENTRE OF GRAVITY. 235

CORRECTION FOR POSITION OF CENTRE OF GRAVITY.-It is

to be noted that an ordinary curve of stability obtained by simply trans-
ferring distances from the cross-curve diagram, as described above, will truly
represent the stability condition only if the centre of gravity, at the particular
displacement, is coincident with that used in constructing the cross-curve
diagram. This, of course, would not usually be the case, and, as already
indicated, a correction must be made.

If the true position of the centre of gravity G be below the assumed one
£?! (see fig. 215), the ordinary curve of stability, as transferred, must be in-
creased throughout by the amount G G x sin. ft if the curve shows righting
arms, and by W x G G\ sin. Q foot tons, if righting moments. Should the
true position of the centre of gravity be above the assumed one, the pro-
cess of correction would be the same, except that it would now, in each
case, be a deduction. Assuming, for illustration, 6\ to be 6 inches above G,
in the case represented by fig. 215, the stability curve would take the
amended form B (fig. 214).

CALCULATIONS FOR CROSS CURVES OF STABILITY.— These are
very simple and may be briefly described : — First, the body plan is prepared
with transverse sections showing the form of the vessel at regular intervals,
as in ordinary displacement calculations. It is an advantage to have the
fore-and-after bodies drawn on each side, and this is sometimes done, although
a separate drawing is frequently used for each body (see fig. 216). Next,
tangent to the midship section, a base line is drawn inclined as required to
the middle line of the vessel, and above the base, waterplanes are intro-
duced at sufficiently close intervals to take account of the vessel's form,
and to suit Simpson's Rule, these waterplanes intersecting the middle-line
plane in parallel lines ; usually an intermediate waterplane is introduced be-
tween the base line and the first waterplane, but this is not shown in
fig. 216. As the deck edge is a point of discontinuity, a waterplane should
occur there. Lastly, a position of the centre of gravity is assumed, and a
vertical line drawn through it to form the axis of the stability moments.
This position of the centre of gravity remains constant throughout all the
cross-curve stability calculations.

The next step is to find the area of each waterplane and the transverse
position of its centre of gravity from the chosen axis. It should be noted
that a plane which does not cut the middle line has to be specially laid
off and its area and centre of gravity determined.

To calculate the transverse position of the centre of gravity of a water-
plane, which is not symmetrical about the middle line as in the present
case, it is convenient to follow the method described for finding the trans-
verse position of the centre of gravity of a half waterplane (see page 29),
that is, taking one side of the plane at a time, put the ordinates and the
ordinates squared separately through Simpson's Rule, and add the products
in each case. Deal in the same way with the ordinates on the other side.
Subtract one total of functions of squares from the other; divide by 2, and

236

SHIP CONSTRUCTION AND CALCULATIONS.

again by the sum of the totals of the products of the ordinates by their
respective multipliers. The result is the distance of the centre of gravity
from the line of intersection of the middle-line plane with the waterplane in
question. All the planes are treated in this way, and the areas and trans-
verse positions of the centres of gravity thus obtained, are combined as
shown in the subjoined table to obtain : —

(i) The displacement below each of the waterplanes.

(2) The horizontal distance of the vertical through the centre of buoy-
ancy corresponding to the displacement below each waterplane from
the vertical through the centre of gravity.

Referring to the table, if H is the sum of the products of the water-
plane areas with their respective multipliers up to any waterplane, and M the
sum of these products multiplied in each case by the distance of the centre

Areas of

Waterplanes.

Simpson's
Multipliers.

Functions for
Volumes.

Distances of

C.G. of Planes

from Axis.

Products for

Moments about

Axis.

An
A,

etc.

2 Av,

iM.'
etc.

etc.

2 ^ljtflj

i J- A 2 d 2
etc.

of gravity of the plane from the chosen axis, then, h being the common
interval between the planes, we have, up to the chosen waterplane —

Displacement in tons — H x — x —

F 3 35

Distance in feet of centre of gravity^ M
from axis / H '

This work is performed for a sufficient number of displacements, and
the results are plotted to obtain the cross-curve of stability at the °iven
inclination, the displacements forming the abscissas, and the horizonral dis-
tances of the centre of buoyancy from the axis, the ordinates. For other
inclinations the process is the same. There are other methods of obtaining
the data required for constructing cross-curves of stability. One of these
consists in the use of a modification of Barnes' tables. For a description
of this method the student is referred to Attwood's Theoretical Naval
Architecture.

THE MECHANICAL INTEGRATOR.— Although cross-curves of stability
can be correctly derived by arithmetical processes as above described, it is
now customary to employ a Mechanical Integrator for this purpose, Amsler's
being the one in common use. This instrument, which has several little
wheels which run on the paper, also a long arm with a pointer, is placed

THE MECHANICAL INTEGRATOR.

237

in reference to the drawing in such a way that the movement of the pointer
round the various sections of the body plan causes readings to be indicated
on dials associated with the little wheels, from which, when affected by certain
multipliers, the displacement and the moment of the displacement about a
chosen axis may be derived. To find the distance of the centre of buoy-
ancy from the axis it is only necessary to divide the moment by the dis-
placement. Full descriptions of the work of calculating the stability in this
way are provided in Reed's Stability of Ships, and Attwood's Theoretical
Naval Architecture, and to these the student is referred for further infor-
mation.

CAUSES WHICH INFLUENCE THE FORMS OF STABILITY CURVES-
BEAM. — We saw in a previous chapter that beam is the element in design
most powerfully affecting the height of the transverse metacentre above the
centre of buoyancy; that, in fact, the height in similar vessels varies as the
square of the beams. It is, therefore, clear that beam will also intimately
affect the forms of stability curves, particularly at initial angles. This may
be shown very simply, as follows. By the metacentric method we have —
Righting arm at inclination Q = G M x Sin. 9,

Fig. 217.

6t *0

O*0*«XS Q9 INCLINATION

from which it is seen that the lever of stability varies directly as the metacentric
height, and that, therefore, a broad vessel with a high metacentre and a large
value of GM will have a stability curve initially steeper than a narrow one,
and vice versa. Using the box form for purposes of illustration, we show in
fig. 217 the actual effect on the whole stability curve of adding to the
beam. Curve A is for a vessel 100' x 20' x 20', floating at 15 feet draught,
with the centre of gravity at 9 feet above the base. Curve B is for a
vessel 100' x 30' x 20', the particulars as to draught and position of centre
of gravity remaining as before. As expected, the latter curve is seen to be
steeper and to have enhanced values of righting arms, although these advan-
tages are associated with a shortening of range. This shortening of range
becomes more pronounced as the vessel increases in beam ; curve G, for
instance, corresponds to a vessel of 35 feet beam, with other particulars the
same as the preceding cases.

INFLUENCE OF FREEBOARD.— Another important element affecting
stability is freeboard, i.e., clear height between load-waterplane and top-deck.

238

SHIP CONSTRUCTION AND CALCULATIONS.

The box-shaped vessels depicted in figs. 218 and 219 are of the same breadth
and draughts, but fig. 219 has greater freeboard. The stability curves for these
vessels are depicted at and D in fig. 220, and it is seen that, up to the angle
at which the deck edge in the one with low freeboard becomes immersed,

Fig. 218.

they are identical. At this point the curve for the shallow vessel receives
a check, and as the deck enters the water, quickly reaches a maximum and
begins to fall. The curve of the other vessel, however, continues to rise
rapidly with further inclination. The explanation is simple, if indeed not

Fig. 219.

quite obvious. In the figures the vessels are shown inclined to the angle
which just brings the high freeboard vessel's deck awash. The deck edge
of the other one is, of course, considerably under water. The points g^ g. 2
mark the centres of buoyancy of the wedges of immersion and emersion in

INFLUENCE OF FREEBOARD.

239

the shallow vessel, and the points g a g 4 the corresponding centres in the
other. Clearly, the distance g 1 g i is less than g 3 g 4 , and, as the volumes of
the wedges are greater in fig. 219 than in fig. 218,

Oi x g 1 g 2 <u 2 x g 3 g,

where v 1 and u 2 are the volumes of the wedges in each case. As the
quantities in this equation are measures of the stability, the relation between
the curves at this inclination is apparent. Beyond this angle both deck
edges are under the surface and the centres g 3 g 4 begin to approach each
other, but much more slowly than the centres gig 2 ; the actual differences in
the values of the righting arms of the two vessels are shown in fig. 220. It
should be stated that the centre of gravity is assumed at the same height
in each case; in each case also the breadth is 35 feet, and draught
15 feet, while one has 5 feet freeboard (curve C)* and the other 10 feet
(curve D)

Fig. 220.

•p

1
1

s£

fc^^i—

o/(

"*" 1

^^L

l^X,

cL/

1 ^

P^-

) !

^r 1

1
1

-5

N. 1
TV '

4&** 1

l
1

•1

1
1

1 I

■0

*K>

90 to

OL4RU.S OF "icli«4iidm

In this comparison we see at a glance the enormous influence of in-
creased freeboard in augmenting the righting levers and extending the
range of stability curves. In the low freeboard vessel the lever is a
maximum at an inclination of 45 degrees; in the other, this is not at-
tained until an inclination of 60 degrees is reached, the value of the
righting lever at this angle being more than double the maximum in the
other case.

Comparing figs. 217 and 220, we note that increased freeboard has a
more beneficial effect on the forms of stability curves than increased beam.
While both elements increase the righting arms, the former also extends
the range which the latter rather tends to curtail.

In the foregoing comparisons we have assumed the centre of gravity
to be at the same height throughout. AVhere the breadth alone is affected
this assumption is fair enough, but where the depth is increased, as in the
high freeboard vessel, a change in the position of the centre of gravity
must be assumed to take place, in the direction of raising it. For a
useful comparison, therefore, the stability curve should be modified to some

240

SHIP CONSTRUCTION AND CALCULATIONS.

extent. If we assume the centre of gravity to be raised 2 feet by the
5 feet addition to the depth, the corresponding stability curve will be that
marked E (fig. 220). A considerable reduction is seen to have taken place
in the lengths of the righting levers. These are, however, except at initial
angles, still considerably greater than those of the low freeboard vessel with
the lower centre of gravity. The range is also greater.

CHANGE IN HEIGHT OF THE CENTRE OF GRAVITY.— The powerful
influence of raising the centre of gravity is manifest from the last illustra-
tion. To a great extent the position of this point is dependent upon the
nature of the stowage. A shipmaster may therefore often make the stability
of his vessel what he pleases. If he finds that she is deficient in stability,
he cannot correct the defect by increasing the beam or the freeboard, but
he can, it may be, stow the heavy weights low down in the hold, and
the light ones higher up, and, by thus lowering the centre of gravity,
attain the same end.

Fig, 221.

,_ — *x

* . 1 *—

^ 1

\
1

1 ^^

1 ^v

1
1

Z-^^lZ* '

9>~ 1 1

10 O* 3* ■**

TO fO

"too

OC^BS-ai or i-.cLtfiAT.or>

It may happen that the stowing of a vessel needs to be conducted
with a view to attaining a high position of centre of gravity. Such would
be called for in a broad, shallow vessel with a consequent high meta-
centre. A low centre of gravity in this case would mean a very large
metacentric height, and this, as we shall see in the chapter on rolling, is
by no means desirable. To get a high centre of gravity, the heavy
portions of cargo should, of course, be stowed high up in the holds or in
the 'tween decks, and the light cargo low down. To illustrate the effect
on the forms of stability curves of raising the centre of gravity, we have
performed this operation on three different box-shaped vessels, and the re-
sulting curves, with those from which they have been deduced, are ex-
hibited in fig. 221. A,B,C, are the original curves of stability; A, l B, ] 0\ the
curves as affected by a rise of 2 feet in the position of the centre of
gravity in each case. Comment on the diagram is unnecessary.

CURVES FOR VESSELS OF NORMAL FORM.— In showing the modi-
fications in curves of stability due to variations in beam, freeboard, and

CURVES FOR VESSELS OF NORMAL FORM,

241

position of centre of gravity, we have cited cases of box-shaped vessels
only; vessels of ordinary form are, however, affected similarly. In fig. 222,
for example, the effect due to beam and freeboard is shown.* A represents
the stability curve of a cargo steamer whose dimensions are — length, 289*5
feet ; breadth, 32*1 feet ; depth, moulded, 23*1 feet. This vessel was as-

Fig, 222,

sumed to have 300 tons of coal in bunkers, and to be laden with a homo-
geneous cargo which completely filled the holds and 'tween decks and
brought her down to her load-waterline. She had a regulation freeboard of
4 feet 7 inches. Curve B shows the effect of reducing the breadth by
2 feet, the conditions of lading being as before, The righting power, as

Fig. 223.

~5o So Co

OEUl«££» OF INCLINATION

in the case of the box vessel, is seen to be much reduced by this altera-
tion.

Curve G illustrates the stability curve of the same vessel after an in-
crease of 6 inches in the freeboard, the density of the cargo being kept

See Reed's "Stability of Ships," p. 104.
Q

242

SHIP CONSTRUCTION AND CALCULATIONS.

as before, and the surplus space assumed to be at the ends of the 'tween
decks.

Further illustrations of stability curves of actual ships are depicted in fig.
223, the particulars of the vessels being given in the table below. Consider-
able variation in stability is here shown. Compare, for example, curves 1
and 4. The first may be considered an example of excessive stability ;
the other of deficiency in this respect. This was borne out when the
vessels were on service. During the voyages made by each when in the
conditions stated in the table,* No. 1 vessel met severe weather and

Description.

Three-deck vessel -
Raised quarter-deck
vessel

Spar-deck vessel
Three-deck vessel,
flush, except for
small forecastle

Quarter-deck vessel
Quarter-deck type,

erections § length
Shelter-deck vessel

modern type
Shelter, deck vessel,

modern type

feet.

ft. ins.

28 5

245

-45

245

i35

210

220

470

Depth.

Meta-
centric
Height

Free-
boaid.

Displace-
ment.

feet.
24.5 REG.D

3*5

tt. ins.

6 6

tons.
380O

17*6 „
24*0 „

i'5
■8

3
8

235°

273O

23'° »

ft. ins.

II 10 M.D.

7
'75

1 4

35°°
612

17 ° ,1

■92

2 1*

2300

34 10 „

■35

7 9

15800

34 10 „

■2

7 9

15800

Cargo.

Pig iron.

Coal or grain.
Coal or grain.

Coal or grain.
Coal.

Coal.

Coal or,grain.

General.

rolled so much that the hull was found to be considerably strained. Had
her initial stability been more moderate, her behaviour would have been better,
and there would probably have been little or no straining. No. 4 vessel,
on the other hand, is a type of many that were lost when making voyages
laden with coal or grain. The elements conducing to deficient stability in
this case are the small metacentric height, low freeboard and almost flush
deck, the only erection being a short forecastle. Vessels Nos. 2 and 3 are
of about the same dimensions as No. 4, but they have certain important
differences which account for their improved stability. No. 2, for instance,
although she has less freeboard than No. 4 to the main deck, has a long
raised quarter-deck erection, which increases the reserve buoyancy ; her meta-
centric height is also greater. No. 3 vessel has double the freeboard of
No. 4, and we are already familiar with the effect of this element on the
stability.

Vessels Nos. 5 and 6 are additional examples of weakness in stability.
No. 5, indeed, was lost on her maiden voyage when in the condition given
above. Her behaviour after leaving port showed her to be very tender.

* Examples 1, 2, 3 ant] 4 are from a paper by the late Mr. Martell in the T.I.N.A. for 1

8S0.

EXAMPLES OF STABILITY CURVES.

243

She first listed to one side, remained in that position for some time, then
returned to the upright, but almost immediately lolled over to the other side.
She continued for some time to heel from one side to the other in this
manner, until eventually she was struck by a sea which caused her to heel
further over, from which position she could not right herself. The water
then poured into the engine-room through the casing door, and the vessel
went down by the stern.* The curve of No. 6 is not so bad as that of
No. 5. The maximum lever is '51 feet, and the range is seen to be 78
degrees. But the vessel was always considered very tender and had to be
handled with care.

Curves Nos. 7 and 8 exhibit two stability conditions of a large modern
cargo vessel of good freeboard. No. 8 is particularly interesting, as it shows
that, with so small a metacentric height as '2 feet, there may be associated
a stability curve quite satisfactory as to range and lengths of righting arms.
In some instances, indeed, vessels having curves of stability of large area
and range have had negative metacentric heights. Such vessels are unstable

Fig. 224.

ntr

30 40 50 60

DEGREE* OF INCLINATION

£0

in the upright position and loll over to one side or the other. In the
case of vessel No. 8, if the centre of gravity were raised 6 inches, the meta-
centric height would be - '3 feet, and the vessel would be unstable from the
upright to an angle of 13 degrees, her stability curve lying below the base
line. She would thus loll over to this angle for her position of equilibrium.
Beyond 13 degrees the curve would rise above the base line, the maxi-
mum lever reaching 1*95 feet at an angle of 53 degrees, and the range 81
degrees. The curve -is depicted in fig. 224, which shows the vessel to have
considerable reserve of stability. In such a case as this, a vessel's ultimate
safety obviously does not depend on there being a positive metacentric height
(see p. 194). The latter, however, is necessary in order that she shall float
upright and not be too easily inclined by the action of external forces. In
the present instance, to gain a positive G M, if the vessel were in quiet
water, the centre of gravity might be lowered by filling a compartment of
the double bottom, but it may be pointed out that it would not be proper
to do such a thing in all cases of instability in the upright condition. It

* See an interesting paper by Mr. Pescod on "Stability of Small Steamers," read before
the North-East Coast Institution of Engineers and Shipbuilders, in 1903,

244

SHIP CONSTRUCTION AND CALCULATIONS.

would have been improper in the case of vessel No. 5 for instance, when in
the condition described above, and would probably have hastened the disaster
which eventually came upon her (see also chapter on Loading and Ballasting).
SAFE CURVE OF STABILITY.— The curves of vessels Nos. 4, 5 and
6, which show unsafe conditions of stability, also cause the question naturally
to arise, "What does constitute a safe minimum curve of stability ? " In
attempting an answer for any given case, two things are to be chiefly borne
in mind — the size of the vessel and the nature of her cargo.

Fig. 225.

u (.0

■

1 ""^t^-_

Z —

■<?-

_— -■ ■**T - ' - ^ *• '

1 **■ ' ^**** s *^*w

j-- »*^***^ 1 **-i

^*^T — ■ *'

I'll ' ^^""n,^

r*""^ \ *

. 1 1 ! ^^

SO *0 So

DEARE.E.S 9F INCLINATION

From the relation : —

Righting moment = displacement x GZ
we make the deduction that vessels of small displacement will be more
affected by movements of weights on deck or water in holds, by shifting of
cargo or by the action of wind or waves, than those of large displacement
with the same curve of righting arms, and that, therefore, a curve of righting
arms suitable enough for a very large vessel, may be quite inadequate for a
very small one.

Again, vessels intended chiefly to load bulk cargoes, such as grain, which
are liable to shift in heavy weather, should have a margin of righting power
in excess of the safe minimum limit Many cargo vessels have to take all

Fig. 226.

AO «© feO

OC&*eE3 OP INCHNATIQN

kinds of cargoes, and for them it is not easy to decide upon a minimum
curve of stability. A well-known firm of shipbuilders, however, have fixed upon
a curve for ordinary medium -sized cargo steamers, of which fig. 225 is a
copy, and experience has shown that vessels so provided are safe and com-
fortable sea boats. Referring to this figure, it will be observed that the
righting arm at 30 degrees and 45 degrees is, in each case, '8 feet, and
that there is a range of 70 degrees.

In fig. 226 we have reproduced the stability curves Nos. 5 and 6 (fig.
223), and have shown in 5 A and 6 A the corresponding safe minimum

DYNAMICAL STABILITY.

245

curves, using fig. 225 as a basis. To obtain a minimum righting arm of *8
feet at 30 degrees in these cases, the centre of gravity in No. 5 would re-
quire to be lowered 1*04 feet, and in No. 6, -5 foot, making the metacentric
heights 179 feet and 1*42 feet, respectively. It will be noticed that the righting
arm at 45 degrees, and the range in each case, exceed those of the standard
curve (see fig. 225). These curves are for small vessels, and, for reasons
already given, we do not say that even the modified curves leave nothing
to be desired ; the stability conditions, however, exhibited by them are a
great improvement on those of the original curves.

DYNAMICAL STABILITY.— The dynamical stability of a vessel at any
angle is the work done in inclining her from the upright to that angle.
It should be carefully distinguished from statical stability, which is the moment
supporting the vessel at the given inclination and tending to return her to
the original position. In heeling a vessel, work is done as follows : —

(1) In raising the centre of gravity.

(2) In depressing the centre of buoyancy.

(3) In creating waves and eddies.

(4) In surface friction.

Fig. 227.

Fig. 228.

Comparing figs. 227 and 228, which show a vessel upright and inclined,
respectively, we note the movement of the centres of gravity and buoyancy,
the former point being obviously nearer the load-waterplane, and the latter
further from it, in fig. 228 than in fig. 227, Items (1) and (2) constitute
the dynamical stability as usually calculated ; items (3) and (4) cannot, from
the nature of the case, be correctly estimated, and in practice are therefore
ignored. The result, however, is on the safe side.

The quickest way of obtaining the dynamical stability is by means of
a curve of moments of statical stability ; for, as shall be shown presently,
the area of such a diagram from the origin to any angle truly represents
the work done on the vessel in inclining her to that angle, omitting, of
course, the effect of surface friction, wave and eddy-making resistances.

As a preliminary, consider the following : — If a force F lbs. acting on

246

SHIP CONSTRUCTION AND CALCULATIONS.

a body causes it to move in any direction through a distance h feet, then,
works on mechanics tell us that the —

Work done on the body — F x h foot lbs.
For example, if F be 10 lbs. and h 5 feet —

Work done = 10 x 5 — 50 foot lbs.
If the force moves along a curved path, the length of the path traversed
is employed in calculating the work done. Consider now a case of two
equal parallel forces acting on a body free to turn. The body will revolve
about an axis passing through its centre of gravity. If the points of ap-
plication of the forces be fixed, the latter will move with the body, and in
turning it through any angle (see fig. 229) if the forces be in lbs. and
the distances in feet.

Work done = (P x A B) + (P x D)
= P{AB + CD) foot lbs.
Since A B = A x Q, and C D = x Q, being the circular measure of the
angle, we may write —

Fig. 229.

Work done = P(A0 + 0) 6

= P x A x foot lbs.,
that is, the work done up to any angle is given by the product of the turn-
ing couple and the circular measure of the angle.

Applying these principles to the case of a ship, let a vessel be assumed
floating at rest in stable equilibrium, then let an external heeling couple
be supposed to act upon her. If, starting at zero with the vessel upright,
this heeling couple be assumed to grow so as always to be equal to the
righting couple, the righting moment diagram at any point will represent the
value of the heeling or upsetting couple at the same point, and, generally,
the curve of righting moments will also represent the curve of upsetting
moments.

Reverting to fig. 214, curve A shows an ordinary curve of righting
moments. Consider an ordinate at 30 degrees, say. On the above assump-

DYNAMICAL STABILITY.

247

tion, it gives the value of the upsetting couple at that angle. Let now
the vessel be further inclined through one degree. If the upsetting couple
be assumed constant through this small inclination, the work done by it
will be equal to the ordinate at 30 degrees multiplied by the circular measure
of one degree. If the base line of the righting moment diagram be in
terms of circular measure, and a line parallel to the base be drawn through
the top of the ordinate at 30 degrees, the work done by the upsetting
couple will be represented by the area of the little rectangle thus enclosed.
At 31 degrees, if the upsetting moment be assumed augmented so as to
equal the righting moment at that angle, and to remain constant while
the vessel is heeled through one degree, the work done will be represented
by the little rectangle between 31 and 32 degrees, the base line, and a
line parallel to the base through the ordinate at 31 degrees.

Proceeding thus by intervals of one degree, the work done by the up-
setting moment from the origin to any angle H will be represented by
the area OHK less the sum of the little triangles between the curve and
the tops of the little rectangles (those indicated in fig. 214 are shown in
black). But by making the intervals infinitely small, the difference between
the area OHK, and the work done by the upsetting couple, is made
infinitely small. We may, therefore, ultimately say, that on inclining the
vessel through the angle OH, the work done, or dynamical stability, is
truly represented by the area of the curve of statical stability from the
origin up to that angle. An ordinary curve of stability thus assumes a
new importance, since, besides showing the variation in the statical righting
moment from point to point, as the vessel is inclined from the upright, it
also measures the amount of energy that must be expended to incline her.

Practical Example. — Assuming the values of the righting moments in
curve A, fig. 214, at intervals of 15 degrees, to be o, 1200, 4900, 9500,
and 8000 foot tons, respectively, what is the dynamical stability at 60
degrees? This is merely a case of obtaining the area of curve A from the
origin to an ordinate at 60 degrees by means of Simpson's Rules. It is
convenient to arrange the figures as follows : —

Degrees of
Inclination.

Righting

Moments in

foot tons.

S.M.

Functions

of
Moments.

O
15

O
I200
4900
9500
8000

4
2

4
1

480O

9800

380OO

80OO

60600

Dynamical stability at 60^ r , -2618

, . ,. .. ^ = 6o6oox = 5288 foot tons,

degrees inclination ) 3 '

•2618 being the circular measure at 15 .

Curve 0> fig. 214, shows the complete curve of dynamical stability for

this vessel.

248

SHIP CONSTRUCTION AND CALCULATIONS.

A knowledge of the dynamical stability* is particularly useful in the
case of sailing-ships as a guide in fixing the area and distribution of the
sails. In such estimates, the sails are assumed braced to a fore-and-aft
plane and the pressure to act dead upon them.

Let fig. 229 represent a sailing-vessel heeled to, and held steadily at,
an angle Q. Here the upsetting and righting moments are obviously equal.
That is,

Pxh = WxGM xSin ft

P being the total wind pressure in tons at the angle 0, h the vertical distance

Fig. 229.

*The dynamical stability may also be obtained by an equation known as Moseley's
formula. Simply stated, this consists in multiplying the vertical movement between the centre of
buoyancy and centre of gravity during the inclination by the weight of the vessel. Reverting
to figs. 227 and 22S, B G is the distance between the centres when the vessel is upright ; B Z
the distance when she is inclined. Therefore —
Work done in inclining"!

vessel to given angle/ = ^^ ~BG)W foot tons.
Now, B V Z -B X R + RZ, and RZ = BG cos d. To find B 1 R we must multiply the volume
(f) of the -wedge of displacement transferred across the ship by the vertical travel of its centre
of gravity i.e., g l h l + g 2 h 2 , and divide by the volume of displacement (V). Thus,

B 1 R = y(g 1 h l + g z h 2 ).

Substituting in first equation, we get —

Work done in foot tons = W\^y{g l h l + g 2 h 2 ) + B G cos 6 - B g)

= ^(yteA + MJ- A 6(i -cos. *))

DYNAMICAL STABILITY OF SAILING SHIPS. 249

in feet at the same inclination between the centre of effort, or centre of gravity
of the sail area, and the centre of lateral resistance, the point through which
the resultant fluid pressure is taken to act, W the displacement in tons^ and
GM the metacentric height in feet. It may be mentioned that the centre of
lateral resistance is usually assumed to be at mid draught.

From the above formula, the magnitude of the angle to which a sail-
ing-ship will heel under a given sail area and wind pressure is seen to
vary inversely as GM, and as it is desirable to have the deck as level as
possible, the importance of a large G M in this class of vessel is apparent.
In medium-sized sailing-ships it should not be less than 3 to 3J feet. In
any special case, the best way of showing the effect of the wind pressure
on the stability is to draw a curve of upsetting moments due to the wind
pressure on the same diagram as the corresponding curve of stability.
This may be done as follows : First, the upsetting moment in the upright
position is calculated, this being equal to the total wind pressure on the
sails in tons multiplied by the vertical distance in feet between the centre
of effort and centre of lateral resistance. Then the upsetting moment acting
when the vessel is inclined to the upright is obtained. In this case, the
effective sail area, and therefore, the effective total wind pressure, are re-
duced, being equal to the values corresponding to the upright position
multiplied by the cosine of the angle of inclination; also the effective
leverage is equal to the leverage in the previous case multiplied by the
cosine of the angle of inclination. Thus, at any angle Q —

The upsetting moment = { U ^> « in x cos. 6 foot tons.
Employing symbols, let —

A = Total sail area in square feet.

H — Distance in feet between centre of effort and centre of lateral

resistance (vessel upright).
/) = Wind pressure per square foot in lbs.
Then, with vessel inclined at some angle 0,
Effective sail area = A cos. Q.
Effective lever = H cos. 6.
Upsetting Moment = A cos. x H cos. Q x p
= A H p cos. 2 Q.
In calculating the steady angle of heel for a vessel under full sail, it
is usual to assume a wind-pressure of i lb. per square foot, which is taken to
be the force of the wind in a fresh breeze. In estimating the effect of a
squall, however, a much larger wind-pressure is assumed.

Values of upsetting moments obtained in this way are set off at corres-
ponding points on the base line of the stability diagram, and a "wind
curve " drawn.

EFFECT OF A SQUALL.— In fig. 230, OBD is an ordinary stability
curve, and A BG F a curve of upsetting moments due to the wind pressure
or wind curve constructed in the way described. Referring to this diagram

25°

SHIP CONSTRUCTION AND CALCULATIONS.

it will be observed that at an inclination P the upsetting and righting
moments are equal. This tells us that but for the energy which the vessel
has stored up in virtue of the unresisted moment area B A, she would be
held at the inclination P. As it is, she passes beyond P to some angle
OH, when the energy of motion is overcome by the stability reserve BOD.
H is approximately twice P, thus a sudden squall striking the vessel

Fig. 230.

. TONS

^B/$$^

7000

111

eooo

Sooo

400s

1
1

p !

3000

C ' • \

$000

1000

H | [

OtljRECS CF INCLINATION

when upright causes her to heel to about twice the angle that a steady
force gradually applied would heel her to.

The wind curve frequently crosses the stability curve at two points,
such as B and F. The portion of the stability curve below the line A F
is absorbed by the upsetting force, while the portion above the line 5 namely,
B D F, is called the reserve of dynamical stability, and as we have seen, is
available to overcome any energy of motion which the vessel may have

Fig. 231.

when she reaches the inclination at which the wind and stability curves
have equal ordinates.

In the case of a sailing-ship rolling freely at sea under full sail, probably
the greatest demand will be made on the reserve of dynamical stability
when a squall strikes her as she is about to return to the upright after
completing a roll to windward.

DYNAMICAL STABILITY OF SAILING SHIPS. 251

Fig. 2 3 1 illustrates this condition. The stability curve above the line
refers to inclinations on one side of the upright ; that below the line to
those on the other side. The inclination to windward is represented by
A. At this point there is a righting lever A B tending to return the vessel
to the upright position; and as the energy which heeled her thus has been
expended, the influence of the righting moment is about to be felt. At this
instant the squall is supposed to strike the vessel. AC D EG F is the wind
curve, and the returning moment becomes suddenly increased from A B to B C,
causing the vessel to return rapidly to the upright, her angular velocity continu-
ing to increase until the angle corresponding to the point E on the other side
of the upright is reached, when it is a maximum. Beyond this, the righting
moment is in excess of the upsetting moment, and as the vessel becomes further
inclined to leeward, her kinetic energy and angular velocity gradually de-
crease, the vessel coming finally to rest at some angle //, when the excess
of upsetting moment, represented by the area B £, is absorbed by the excess
of dynamical stability EKME. Her energy of motion being now expended,
the vessel begins to return by virtue of her excess of righting moment and,
if the wind curve be assumed to remain as before, she will oscillate for a
little about the angle corresponding to the point E and finally come to rest
at that angle.

We have neglected the retarding effect of the friction of the water and
the hull surface, and of such head resistances as bilge keels. These con-
siderably reduce the inclinations to which the wind heels the vessel, and if
the wind were suddenly to fall, would, with the assistance of the air re-
sistance on the sails, gradually bring her to rest.

QUESTIONS ON CHAPTER IX.

1. What is a curve of statical stability? How is such <% curve usually drawn?

2. Distinguish between the terms "righting arm" and "righting moment." A vessel has
a displacement of 4000 tons and a metacentric height of 2 feet ; what are the values of the
righting arm and righting moment when the vessel is heeled to an inclination of 10 degrees ?

Arts.— R.A. = -348 foot; R.M.= 1392 foot tons.

3. In the case of vessels of ordinary form, correct values of righting arm and righting
moment cannot be obtained at large inclinations by using the metacentric height ; explain
why. For what special form of vessel is the metacentric height method correct ?

4. Construct <x curve of righting arms for a vessel of cylindrical section, 15 feet in
diameter, floating with its axis in the waterplane, the centre of gravity being iS inches below
the middle of the section.

5. A vessel's metacentric height is 2 feet 6 inches ; show how you would construct
the tangent to the curve of righting arms at the origin, and prove that your method is
correct in principle.

6. Show that in a submarine vessel floating below the surface, the centre of buoyancy and
metacentre coincide with the centre of bulk, and explain what in this special case are the con-
ditions of equilibrium.

252 SHIP CONSTRUCTION AND CALCULATIONS.

7. If, in the previous case, the centre of gravity be below the centre of bulk, show that
if the vessel be turned about a horizontal axis passing through the centre of bulk, the curve of
stability will be the same whatever be the direction of the axis.

8. Prove Atwood's formula for the statical stability of a vessel at any angle of heel.
Show how a. statical stability diagram is constructed, and explain its uses.

9. Explain the terms "angle of maximum stability, " and "range," as applied to curves
of stability. A box-shaped vessel 35 feet broad, and 35 feet deep, floats at a level draught
of 17 feet 6 inches. The metacentric height of the vessel being 2 feet, construct the curve
of righting arms, indicating the "angle of maximum stability," the value of the "maximum
righting lever," and the " range."

10. Describe in detail how you would proceed to obtain the statical stability at large
angles of inclination of a vessel of known form.

11. A vessel having a deep forward well ships a heavy sea. Assuming the water ports
to be set up with rust and inoperative, discuss the effect of the filling of the well on the
vessel's stability, and state whether her safety is likely to be thus endangered.

12. In Barnes' Method of calculating the statical stability of a vessel, show clearly how
the "wedge correction" is made. The uncorrected sum of the moments of the wedges of
immersion and of emersion for an inclination of 30 degrees is 263,00x3, and the vessel's dis-
placement is 2000 tons. The volume of the layer is 600 cubic feet, and the horizontal distance
of the centre of gravity of the radial plane at 30 degrees from the intersection with the middle-
line plane is 1*5 feet. Find the value of the righting arm (1) assuming the immersed wedge
in excess and the centre of gravity of the radial plane on the immersed side; (2) assuming the
immersed wedge in excess and the centre of gravity of the radial plane on the emerged side ;
(3) assuming the emerged wedge in excess and the centre of gravity of the radial plane on the
immersed side ; (4) assuming the emerged wedge in excess and the centre of gravity of the
radial plane on the emerged side.

Note. — The centre of gravity of the layer may be assumed to be in the same vertical with
the centre of gravity of the radial plane, and B 6 = 5 feet.

Ans.f^ and M' Ri S htin S Arms = 1*24 feet.
*\(2) and (3), ,, = 1-27 ,,

13. A vessel whose displacement is 3000 tons, has a righting-arm of 1 foot at an in-
clination of 30 . Cargo weighing 300 tons, whose centre of gravity is in the middle line at
a depth of 1 foot below the common centre of gravity, is discharged; and the vertical
through the centre of buoyancy of the layer through which the vessel rises when at an
inclination of 30 is 6 inches on the immersed side of the vertical through the centre of
buoyancy of the vessel corresponding with the load draught at that inclination. Find the
length of the righting arm after the removal of the cargo.

Arts. — fths of a foot.

14. What are cross curves of stability? How are these related to the ordinary stability

curves ?

15. A vessel of constant circular section, 120 feet long and 14 feet in diameter, has its
centre of gravity 1 '5 feet below the axis ; construct to scale cross curves of stability for trans-
verse inclinations of 30 , 60% and 90 .

16. Referring to the previous question, deduce an ordinary curve of stability for the vessel
when floating with its axis in the waterline. If the centre of gravity be 9 inches below the
position assumed in making the cross curves, show how the necessary correction would be
made at the various inclinations, and plot the new curve.

QUESTIONS. 253

17. The plans of a vessel being given, state hilly how you would prepare the body plan
for the calculation of a cross curve of stability.

18. A vessel inclined transversely is cut by a series of horizontal equidistant planes at
intervals of 3 feet, the intersection of the middle-line plane being parallel to the top of keel.
The first plane touches the vessel's bottom tangentially. The areas of the successive planes are
o, 30°» 590) 880, and 1 150 square feet, and the horizontal distances in feet of their respective
centres of gravity from the vertical through the vessel's centre of gravity, omitting the tangent
plane, are 1 "4, *9, '4, and *l, on the immersed side. Construct the cross curve of stability.

19. What are the causes which influence the forms of curves of stability? Give an example
of such curves for

(1) a flush-deck vessel of low freeboard;

(2) the same vessel fitted with a continuous watertight shelter deck.

20. Discuss the comparative effect on curves of stability of increase of breadth and in-
crease of freeboard. Taking a rectangular vessel 100 ft. long, 20 ft. broad, 15 ft. deep, floating
at a level draught of 12 ft,, with the centre of gravity at 7 ft. above the base; show the effect
on the stability curve of

(1) an increase of 4 ft. in beam,

(2) an increase of 4 ft. in freeboard,

the draught and position of centre of gravity remaining the same in each case.

21. What is meant by the dynamical stability of a vessel? In inclining a vessel from the
upright position explain the several ways in which work is done.

22. A rectangular pontoon 100 feet long, 25 feet broad and 25 feet deep, floats in salt-
water at half depth with one of its sides horizontal ; the metacentric height is I foot.
Calculate the dynamical stability at an angle of 45 .

Ans. — 487 foot tons.

23. Prove that the work done in inclining a vessel from the upright to any angle, is equal
to the area of the curve of righting moments up to that angle.

24. The ordinates of n curve of righting arms measured at equal angular intervals of io°,
starting from the upright, are — o, *4, 7, -9, and i*o feet. Find the dynamical stability at 40
inclination, the displacement being 2500 tons.

Ans. — 1 102 foot tons.

CHAPTER X.
Rolling.

THE time that a vessel, rolling freely in undisturbed water, takes to com-
plete an oscillation from port to starboard, or vice versa, is called her
period of a single roll. Theoretical investigations in this subject are
based on the assumption that the rolling medium is a perfect or frictionless
fluid, so that in calculating the period of roll, the fact that water offers
substantial resistance to the movement of the vessel is ignored. The result
thus obtained is found to be nearly true, since, from actual rolling experi-
ments, fluid resistance, while greatly limiting the arc of oscillation, appears
to have little influence on the period.

Early investigators were wont to think that if a vessel had great initial
stability, and was, therefore, difficult to move, she would also be steady in a
seaway. They were struck with the apparent analogy between a rolling ship
and an oscillating pendulum, and thought that a ship might be looked upon
as a simple pendulum suspended at the metacentre of length equal to G M,
the distance between the metacentre and the centre of gravity.

Now, the period in seconds of a single swing of a simple pendulum,
from left to right, or vice versa, is

T = 3-1416 J L,

where / is the length of the pendulum, and g the acceleration due to
gravity.

If the above analogy between the pendulum and the ship were correct,
G M might be substituted for / in this formula, and, consequently, the ship's
rolling period would lengthen with increase in G M. We find, however, such
to be by no means the case, all experience going to show that vessels of
small metacentric heights are of longer periods, that is, make fewer rolls per
minute, than those having large metacentric heights. Thus, the assumption
that a ship is a simple pendulum, with its whole weight concentrated at the
centre of gravity, and with a fixed axis of oscillation at the point M, is
clearly an erroneous one.

As a matter of fact, a ship has no fixed axis of oscillation. The in-
stantaneous axis is for most vessels not at M, but in the vicinity of the
centre of gravity, and it is usual to assume it as passing through that point,
While accepting this as a fair approximation, it must not at the same time
be forgotten that the centre of gravity itself, though fixed relatively to the

254

INSTANTANEOUS ROLLING AXIS. 255

ship, really describes a path in space as the vessel rolls. To obtain the
instantaneous axis we may proceed as follows : —

Referring to fig. 232, let W L be the waterplane, F F the curve of
flotation, i.e., a section of the surface tangent to the various waterplanes
which cut off a constant displacement as the vessel rolls, and G the centre
of gravity. Now, neglecting the presence of the ship, assume the surface of
flotation and the level water surface to become solid, and the former to roll
or slip without friction along the latter as the vessel oscillates. F, the point
of contact of the surfaces FF and W L, is a point in the oscillating vessel,
and will move, at any instant, about a centre somewhere in the line F 0,
Another determinable point in the vessel is the centre of gravity. It has
vertical motion only, since the forces acting when the vessel is rolling freely

Fig. 232.

are purely vertical , therefore, the centre about which G turns is in the line
G 0. The axis of the vessel at the instant considered is obviously at 0,
the point of intersection of the lines FO and GO. The point in most
cases is near G, so that very little error is introduced by the assumption
that the axis passes through G.

Let us consider what actually takes place when a ship is rolling un-
resistedly in still water. This case is a purely hypothetical one, but it
offers a convenient starting-point.

Suppose a vessel floating freely and at rest is acted upon by an external
force causing her to roll through some angle, say, to port. The work done
is represented by the dynamical stability of the vessel at the angle at which
she comes to rest. She has then energy due to position, which, on removal
of the external force, takes effect in returning her towards the upright.

256 SHIP CONSTRUCTION AND CALCULATIONS.

When the vertical is reached the energy of position becomes transformed
into energy of motion, the vessel attaining a maximum angular velocity.
The energy of motion now carries the vessel to starboard, to the same
angle as that reached on the port side, where she once more regains energy
of position, which, in turn, sends her back to the upright. And so the
rolling goes on, since, by our assumption, there is no external resistance.

The formula for a single roll in the above hypothetical circumstances is—

k
v g m
where T is the time m seconds of a single roll, m the metacentric height in feet,
and g the acceleration due to gravity (= 32*2 feet per second per second).

The symbol k is known as the transverse radius of gyration. What
this quantity is may be explained by stating that if the whole weight of
the oscillating vessel could be concentrated at a point distant k from the
axis of oscillation, the effect would be the same as with the vessel as she
is, that is, the period of oscillation would be the same. To find the
value of k in any case, it is necessary to assume the vessel's weight
divided into very small elements w, say, and to obtain the distance between the
centre of each of these elements and the rolling axis ; then, if r be taken
to represent any of these distances, and W be the total weight of the
vessel —

Sum of all the products w x r 2
h = W '

The numerator of this expression for k 2 is the moment of inertia of the
vessel about the chosen axis ; calling this /

Using the value given for g, the formula for the period may be written —

We now see why vessels having large metacentric heights are of quicker
motion, i.e., shorter period, than those with smaller values, for evidently T
becomes reduced with increase of /??, and vice versa. Also, the period is
increased or decreased with corresponding changes in the value of the radius
of gyration, which varies according to the distribution of the weights on
board the ship, being increased by spreading them out from the centre of
gravity, and decreased by crowding them about that point.

As a practical example, let us obtain the rolling period for a vessel of
3000 tons displacement, whose metacentric height is 2*5 feet, and radius of
gyration 17*16 feet. By substitution we get —

17-16
T = '554 = = 6 seconds.
J 2-5

r= '554 r:

STILL WATER ROLLING PERIOD, 257

Vertical movements of weights have greater influence on the period than
horizontal movements. For instance, in the above vessel, if 120 tons were
raised 14 feet, i.e., from a position 7 feet below the centre of gravity to one 7
feet above it, the period would be increased to 6*8 seconds, while winging out
this weight 14 feet from the centre would only lengthen the period to 6'i seconds.
As well as by calculation, the still-water rolling period may be found experi-
mentally. To do this it is only necessary to set the vessel rolling by some
artificial means, and to note the number of complete rolls she makes in a
certain time. The period of a single roll may then be got by dividing the
time by the number of rolls- This follows from the fact that the time taken
per roll, for all inclinations up to which the curves of statical stability are
straight lines — that is, about 12 to 15 degrees— is the same, a characteristic
known as Isochronism.

It may be well to state here that it is important to know the value
of a vessel's still-water rolling period in order to predict her probable be-
haviour at sea. Vessels seldom roll to dangerous angles in still water, but,
as we shall see presently, may do so among waves, unless precautions have
been taken to provide them with suitable still-water periods.

SEA WAVES, — Before dealing with the rolling of a vessel among waves,
it will be necessary to give some attention to the structure of the latter in
the light of modern theory, in order to obtain a clear idea of the action of
water on a vessel when the surface of the former is undulated into wave shape.

Waves are generated by the action of wind on the sea, and are the
principal agents causing ships to roll. There have been various theories as
to the action of wave water, the most satisfactory of which, and the one now
generally accepted as representing the case best, being that known as the
trochoidal theory.

The groundwork of this theory is, that only the form of the wave travels,
and that the particles of water affected move in small circular orbits about
horizontal axes. That some such action does take place will be obvious to
anyone who observes the movements of a piece of driftwood afloat among waves.
It will be noted that the wood does not travel with the wave, but merely
moves backwards and forwards, showing clearly that the water particles sup-
porting it move only a short distance as the wave passes.

According to this theory, a section of a wave in the plane of the line
of advance, has for its outline a trochoid, i.e., a line described by a point
having uniform circular and linear motion. A rough, but simple way of draw-
ing a trochoid, is as follows : — Take any point between the centre and the
circumference in a circular paper disc, and let the latter be rolled without
slipping along a horizontal line ; the path described by the point is a trochoid.
The reader should try this for himself. The theory also states that originally
horizontal layers below the surface become, when under wave motion, distorted
into trochoidal forms of the same general character as that of the upper sur-
face, and that columns of water originally vertical curve towards the wave
crest. The hollows and crests of the various trochoids are immediately under

258

SHIP CONSTRUCTION AND CALCULATIONS.

each other, and, therefore, the trochoids are all of the same length. But they
become flatter as the depth below the upper surface increases, the particles
moving in smaller and smaller orbits, until finally the wave, form disappears.
Fig. 233, which exhibits in section part of a trochoidal wave, illustrates some
of the points referred to. In this figure the original surfaces and subsurfaces
in still water are shown by dotted horizontal lines, the same surfaces when
in wave form by full curved lines. The orbits of surface and subsurface
particles are also indicated. The lines containing the centres of these
orbits (shown full) are seen to be at higher levels than the corresponding
still-water lines, showing that in the wave, as well as kinetic energy, or energy
of motion, the particles have also potential energy or energy of position.

Another fundamental point in this theory is, that the pressure of a
particle in the wave is affected by the centrifugal force generated by its
orbital motion, and acts normally to the particular surface in which the par-
ticle lies ; so that, as the slope varies at each subsurface, the directions of

Fig. 233.

the normals also continually vary. All this has to be remembered when
considering the resultant of the wave forces which act on a vessel afloat
among waves.

The length of a wave is the horizontal distance from crest to crest, or
hollow to hollow; the height is the vertical distance from hollow to crest;
the period of a wave is the time it takes to move a distance equal to its
own length. From calculations based on the trochoidal theory, we have the
following : —

t, • j r , 2 x V1416 x lensrth / length

Period of wave in seconds = / ^ — 5 — = / -° ,

v 9 v 5-124

Speed of wave in feet per) /length x a , -. —

second

x 3-1416

Results obtained from these formulae are found to agree closely with those
of observations of actual waves. Atlantic storm waves 600 feet in length, for

SEA WAVES. 259

instance, have observed periods of 11 seconds, and from the formulas, using
this length of wave, we get —

Period = / = io'8 seconds.

v 5* I2 4

Speed — / 5*124 x 600 = 55*4 feet per second.

The heights of waves have an important influence on rolling. The magni-
tude of the maximum angle of slope of a wave depends upon the ratio of
the height to the length, and it will be found that the extent of the arc
through which a vessel oscillates, is largely governed by the value of this slope
angle. The heights of well-defined ocean waves are usually found to vary from
"sir to tV °f tne ^ngths, in long and in short waves, respectively, the steepness
of waves decreasing with increase in length.

ROLLING AMONG WAVES.— We have seen that the effect of the
internal wave forces is to cause the resultant buoyant pressure on a surface
water particle to act normally to the wave slope, and it must now be added
that the same forces act upon the weight of any small floating particle or
body, and cause the resultant force also to act normally to the wave slope,
but in a line opposite to that of buoyancy. The truth of this was proved
experimentally by Dr. Froude in the following manner : — Taking a small float
of cork he fitted it with an inclined mast, from the top of which he sus-
pended a simple pendulum. He then placed the float on waves, when it
was observed that the pendulum did not hang vertically but took up a
position perpendicular to the wave slope.

Now, a ship displaces a considerable amount of wave water, and cannot,
properly speaking, be looked upon as a surface particle. It intersects many
subsurfaces, the pressures on the particles of which act normally to the
curves of these subsurfaces, and the resultant pressure, on the whole body,
is normal to a mean subsurface ; but in actual calculations it is usual to
consider the vessel as very small relatively to the wave, and to treat it for
all practical purposes as a surface particle, the resultant lines of pressure and
weight being assumed to act normally to the slope of the upper surface of
the wave.

On this assumption, a vessel among waves will tend to place her masts
parallel to the normal to the wave slope, which virtually becomes her upright
position of equilibrium. This is illustrated in fig. 234, where the centre line,
that is, the line of the masts, is shown inclined to the wave normal, with
a moment W x G Z in operation tending to bring them into parallel lines
and thus place the deck parallel with the wave slope. In calculating the
angle of inclination of the vessel to the vertical at any instant when among
waves, this modified righting moment, which is assumed to be proportional
to the angle between the line of the ship's masts and the wave normal at
that . instant, is employed.

It is not our intention to attempt a description of these calculations,

260

SHIP CONSTRUCTION AND CALCULATIONS.

as they are difficult and would be quite out of place in a work of this
kind. It is easy, however, in general terms, to predict the behaviour of a
vessel among waves when the periods of ship and waves are known.

Where the still water period of a vessel is very short in comparison
with the period of the waves she is among, caused by her being specially
broad and shallow, or having a cargo of great density placed low down in
her holds, she will tend to keep her masts close to the wave normal, as
depicted in fig. 235. Her motions will be quick and jerky, and although
she will generally keep her decks clear of water, she cannot otherwise be

Fig. 234.

considered satisfactory. Her rapid motions are likely to strain the hull,
especially during rough weather, and she will obviously be an uncomfortable
boat in which to traveL

Different results are obtained when the ratio of the periods is reversed,
i.e., when the still water period is very long compared with the wave period.
A vessel so circumstanced will be an easy roller, as may be readily ex-
plained. Assume such a vessel, for example, to be upright when a wave
approaches her. Under the influence of the wave forces, she will immedi-
ately begin to heel, but her period being long compared with that of the
wave, she will not have gone far when the wave normal, having passed

Fig. 235.

through its maximum angle to the vertical, at about the mid-height of the
wave, will have returned to the upright, bringing a crest under the vessel.
As the crest passes, the tendency of the wave pressures in the back slope
will be to arrest the inclination of the vessel, and return her to the upright.
And so the next hollow will find her a little inclined to the other side.
This inclination will, in turn, be arrested by the following wave, and thus
the departure from the upright of such a vessel will be small, and she will
maintain a comparatively level deck. Such a state, of things is highly
desirable in warships to ensure a steady gun platform, and for obvious
reasons it is also sought after in merchant vessels.

ROLLING AMONG WAVES.

261

From the formula—

v m
it is seen that in order to obtain a long rolling period, /f, the radius of
gyration, must be increased, and aw, the metacentric height, must be reduced,
as much as possible. This would mean concentrating the weights away
from the middle line, narrowing the beam, or raising the position of the
centre of gravity. More important considerations than those of rolling limit the
extent to which the foregoing modifications may be carried out. It is impractic-
able, for instance, in merchant vessels to bank the weights against the sides,
although with general cargoes something may be done by judicious stowage,
while to bring down the value of m by reducing the beam or stowing the
weights high in the vessel might seriously affect the stability, and no careful
designer would recklessly do that. Experience must be the guide here, it
being remembered that, generally speaking, vessels of great displacements
may have smaller values of m than those of small displacement.

A critical case arises when the half period of the waves is the same
as the ship's still-water period, and she is rolling broadside on to the former,
a state of things known as synchronism.

In fig. 236 the effect of this coincidence of the effective time of the

Fig. 236.

two periods on a vessel's behaviour when rolling in a frictionless fluid, i.e.,
without resistance, is depicted. Referring to this figure, at A the vessel is
in the hollow, and is supposed to be upright when the wave reaches her.
As she rises on the latter, the internal wave forces cause her to heel from
the upright, and her period agreeing with the half period of the wave, she
reaches the end of a roll at the first wave crest. On the back slope the
wave forces will assist the ordinary statical moment to return her to the
upright, and to a maximum inclination on the other side of the vertical,
which she will reach at the hollow, and which will be greater than if she
had oscillated under her statical moment alone. The wave forces in concert
with her statical moment will again change the direction of motion, and at
the next crest, where she will complete another roll, her maximum inclina-
tion will be further increased. Thus she will continue to roll, reaching
a greater maximum inclination at each crest and hollow, until she finally
upsets.

Theoretical investigations show that the increment of roll due to the
wave impulse is equal to f, or about if times the maximum wave slope. Thus,
if this were 6 degrees, the maximum inclination would be increased each

262 SHIP CONSTRUCTION AND CALCULATIONS.

Lime by 9 degrees, and her arc of oscillation by 18 degrees; so that the
effect of a few such waves would be to put the vessel on her beam ends.

Dr. Froude proved the truth of the foregoing theory by experimenting
with little models in a tank. Waves were generated having a period double
that of the models in still water, and the latter when placed in the tank
were upset after the passage of a few waves.

We thus see that a vessel is most seriously situated when broadside
on to waves whose period of advance is double that of her own still-water
period. In some respects the ship, in receiving the wave impulse as above
described, resembles an oscillating pendulum which has additional force
applied to it periodically at the end of an oscillation in one direction, and
just when it is about to return, the effect of which is to increase the angle
of swing each time. Another illustration is given by a body of soldiers
crossing a bridge, when the period of march keeps time with the period of
vibration of the bridge, a state of things which, continued long enough,
would greatly increase the amplitude of the vibrations, and might eventually
bring the structure down.

Summarising the foregoing remarks and deducing obvious inferences
there from, we note : —

(1) That vessels whose periods are very short in comparison with the
waves, will tend to place their masts parallel to the wave normals ;
that the angular velocity of such vessels may, in stormy weather,
become very great and the rolling heavy; that excessive transverse
straining may thus be developed, with a particular tendency to throw
out the masts.

(2) That vessels of long periods (single roll), if among waves with half
periods considerably less, are likely to be slow rollers, and to incline
through moderate angles from the upright ; that this is a most de-
sirable state of things in both mercantile and war vessels, and after
sufficiently allowing for stability, should be aimed at in new designs.

(3) That vessels having periods which keep time with those of the
waves are badly, if not dangerously, situated ; that such synchronism,
as has been borne out in actual cases of which records are avail-
able, is likely to conduce to heavy rolling and severe transverse
straining.

A practical example of the effect of synchronism is afforded in the case
of H.M.S. Royal Sovereign, a large warship which from her design was ex-
pected to be very steady among waves at sea, and in general proved
herself to be so • but on one occasion, when sailing in company of other
vessels, there being a slight swell on the sea at the time, she rolled con-
siderably, her maximum arc of oscillation reaching to 32 degrees, while the
other vessels,, which were of quicker periods, were but slightly affected.
Observation showed the waves to have a period which synchronised ap-
proximately with the Royal Sovereign's single roll period of 8 seconds. On
another occasion, when broadside on to a series of synchronising waves, she

ROLLING AMONG WAVES. 263

rolled through maximum arcs of 50 to 60 degrees. Excessive rolling is also
reported of another vessel of this class, H.M.S. Resolution, the circumstances
pointing to sychronism between the ship and apparent wave periods.

These examples show how difficult it is to altogether avoid the effects
of synchronism. Actual observation has shown ordinary storm waves to
average 500 to 600 feet in length, with periods of 10 to n seconds, only
in exceptional cases longer waves being met with. Consequently, vessels
having still-water periods of 8 seconds like the Royal Sovereign class should
be expected to practically escape synchronism. Experience, however, has
shown that circumstances may arise which shall cause the unexpected to
happen. But even where there is synchronism, a vessel of long period is
better situated than one of short period. In the former case, the waves
keeping time are longer and less steep and have smaller maximum slope-
angles than in the latter. Thus, the increment added to the angle of roll
at each hollow and crest is less in the vessel of long period than in the
other.

Of course, a master with his vessel well in hand is usually able to help
matters considerably when his vessel is rolling excessively on account of
synchronism. Should the rolling become heaviest when she is lying broad-
side on to the waves, he may disturb the coincidence of the periods by
changing to an oblique course, which would lengthen the effective time of
the wave. Should, however, the synchronism be developed when sailing on
an oblique course, he may affect a cure ' in various ways. He may turn
his vessel into the wave trough, or direct her head to the line of crests,
or if he wishes to keep the original course, he may change the effective
time of the waves by increasing or reducing the speed of the vessel. Thus,
by skilful navigation, much can be done even with a vessel of bad design.

RESISTANCE TO ROLLING.— Although synchronism will always tend
to make rolling heavy, as in the case of the Royal Sovereign, the resistance
due to the friction of the water with the surface of an oscillating vessel,
with that spent in the creation of waves, etc., will minimise the rolling at
all times. Suppose, for instance, a vessel is set rolling in still water, and
that, at a given instant, the external force causing her motion is removed,
thus allowing her to roll freely. Her maximum range of oscillations will
immediately begin to diminish. In any single oscillation, the difference
between the maximum angle, say to starboard, from the following one to
port, will be a measure of the resistance overcome. But when among
waves whose period keeps time with that of her own motion, the periodical
impulse given by the wave will cause the maximum inclination to be in-
creased with each oscillation, so long as the resistance of the water is less
than the increment of force due to the wave impulse. With increase of
angle, however, the speed of oscillation will increase, since vessels describe
the largest arcs in nearly the same time as the smallest. And, since the
resistance of the water increases rapidly with the speed, a range of oscilla-
tions is soon reached, to sustain which the repeated impulse due to syn-

264 Ship construction and calculations.

chronism is necessary. Moreover, although the period when the arcs of
oscillation are large are only slightly greater than when they are small, the
difference is sufficient to disturb the synchronism. The wave impulse does
not occur at the same critical moment each time, and a fraction of the
resistance being thus unbalanced, it takes effect in reducing the angular
velocity, the rolling becoming less heavy. This reduction may increase the
period, and again cause synchronism, with consequent increase in the rolling,
which, as before, will in turn be arrested. We thus see that oscillations
sufficient to place a vessel on her be:im ends, or to overturn her, are
unlikely to occur in a resisting medium such as water.

ANALYSIS OF RESISTANCE.— Many years ago, Dr. Froude m carrying
out experiments on the resistance of vessels to rolling, divided it into three
parts, viz., that due (1) to the hull surface; (2) to keel, bilge-keels, dead-
wood, and the flat parts of the vessel at either end ; (3) to surface disturbance.
He obtained quantitative results by calculating items (1) and (2) from the
plans of the vessels, and placing the difference between the sum of these
items and the actual resistance obtained from experiments to the credit of
item (3). The results showed the hull surface resistance to be less than 2
per cent., and the keel, bilge-keel, and flat surface resistance from 18 to 20
per cent, of the total, leaving about 80 per cent, as due to surface dis-
turbance and the creation of waves.

While it is known that the creation ot a very small wave would be
sufficient to account for this residual resistance, subsequent experiments and
investigations have shown that item (2) has probably been under-estimated.
In making his calculations for the resistance of flat surfaces, Dr. Froude
took i*6 lbs. as a co-efficient of resistance per square foot at a velocity of
one foot per second, and assumed the resistance to vary as the square of
the velocity. This co-efficient he had obtained previously by oscillating a flat
board in deep water. In the case of bilge-keels, it is now pretty well estab-
lished that this figure, taken with the surface area of the bilge keels, does
not represent the extinctive value ot these appendages. On the assumption
that the whole work of extinction, due to the fitting of bilge keels, might
be credited to a virtual increase in the co-efficient of resistance per square
foot of bilge area, and that the resistance varied as the square of the velocity,
Sir Philip Watts pointed out that, in the case of the warships Volage and
Inconstant^ instead of i'6 lbs., the co-efficients at a mean velocity of one
foot per second should be 87 and 7*2 lbs., respectively.

In rolling experiments carried out in 1895 on H.M.S. Revenge^ a war-
ship of large displacement, similar results were obtained, the corresponding
co-efficient being 11 lbs. for a swing of 10 degrees, rising to about 16 lbs.
for a swing of 4 degrees.

Commenting on these results, Sir Wm. Whyte* pointed out that, as well

* See a paper by Sir Wm. Whyte in the Transactions of the Institution of Naval Archi-
tects for 1895.

RESISTANCE TO ROLLING. 265

as offering direct resistance, bilge keels create further resistance by indirectly
influencing the stream-line motions that exist about an oscillating ship.

Investigation* has fully borne this out. Professor Bryan has shown that
the motion of a rolling vessel, particularly if she be of sharp form at the
bilge, gives rise to counter currents in the water which strike the surface
of the bilge-keels and increase their extinctive value ; also, that the presence
of the bilge-keels cause discontinuous motions in the surrounding stream
lines, the result of which is a gradual reduction in the speed of the streams
as they approach the keels and an increase of pressure on the hull surface,
giving rise to turning moments tending to arrest the angular motion of
the ship. Crediting these resistances to the bilge-keel area, Professor Bryan
estimates the effect * of the counter currents as equivalent to doubling Dr.
Froude's co-efficient, and the effect of the discontinuous motion to quadrup-
ling it.

It should be stated that the foregoing analysis is based on the assumption that
the vessel has no forward motion, but rolling motion only. From the results
of rolling experiments! with a destroyer, it is shown that the effect of discon-
tinuous motion is much reduced, when a ship has motion ahead, the diminution
increasing with the speed ahead for the same angle of roll ; the apparent
reason being that the keel surfaces strike the water at an oblique angle,
and thus do not create such masses of dead water as when rolling without
forward motion.

But however the work done by bilge-keels be analysed, the point of most
importance concerning them is that they are invaluable as a means of
reducing rolling. Experience has amply shown this in a general way, but
figures deduced from actual experiments are perhaps more convincing.
H.M.S. Repulse, a large battleship, was, as an experiment, fitted with bilge-
keels 200 feet long and 3 feet deep, and when amongst synchronous waves
at sea, was found to reach only half the maximum angles of oscillation at-
tained by her accompanying sister vessels, which were without bilge-keels. In
the years 1894-5, rolling experiments, with and without bilge-keels, were con-
ducted on the Revenge, a vessel of the same class as the Repulse. In still
water it was found that, starting with an inclination of 6 degrees, without
bilge-keels it took 45 to 50 swings to reduce the angle to 2 degrees, and
with bilge-keels, similar to those on the Repulse, only 8 swings. Again, it
was noticed that, starting at 6 degrees inclination, after 18 swings the vessel
without bilge-keels reached an angle of 3! degrees, and with bilge-keels,
an angle of 1 degree. In the case of the destroyer above referred to,
the decrement of roll for 4 degrees mean angle of roll was, without bilge
keels, '24 degrees, and with bilge-keels, '5 degrees. Most important of all
is perhaps the effect of bilge-keels on rolling when vessels have motion

*See a paper in the Transactions of the Institution of Naval Architects for 1900.
fSee an interesting paper by Mr. A. W. Johns in the Transactions of the Institution
of Naval Architects for 1905.

266 SHIP CONSTRUCTION AND CALCULATIONS.

ahead. In the case of the Revenge^ starting at an angle of 5 degrees from
the vertical in each case, after 4 swings the inclination with no motion
ahead was 2-95 degrees, and at a speed of 12 knots, 2*2 degrees; after 16
swings the corresponding inclinations were 1*15 degrees and '25 degrees.
In the case of the destroyer the resistance to rolling for 4 degrees mean
angle of roll, at 17 knots, was 3 J times greater than when not under weigh.

The reason of the greater extinctive value of bilge-keels in vessels
under weigh is due to their having at each instant new masses of water to
set in motion, and the energy so imparted being continually left behind
and thus lost to the ship.

The experiments with the destroyer brought out another point, viz., that
forward motion tends to reduce the rolling period both with and without bilge
keels. The double period in seconds with no motion ahead was found to be —
with keels, 5 '5 9; without keels, 5 '61. At about 17 knots speed the corresponding
figures were 5*4 and 5*46 respectively. At higher speeds the reduction was
still more marked. Another point to be noted in respect to bilge-keels is
that they are more effective in small quick-rolling vessels than ■ in large vessels
of slow angular motions. This follows because the resistance bilge-keels meet
with from the water increases with their speed of motion through it, and
because the power of this resistance in arresting angular motion is greatest
when the oscillating body to which the bilge-keels are attached is of re-
latively small weight and inertia. The importance of having these appendages
on small vessels of quick-rolling period is thus apparent. The advantages of
bilge-keels are now generally recognised, and they are regularly fitted to both
war and merchant vessels. In warships they are frequently of considerable
depth; in merchant vessels they are seldom more than 12 to 15 inches deep,
and are often less ; but even when so limited in size, their effect on the
behaviour of vessels at sea has been most beneficial.

WATER CHAMBERS.— Besides bilge-keels, various other more or less
successful methods have been advanced for minimising the rolling of ships.
The best known of these consists in having a chamber partially filled with
water fitted across the ship, so that when the vessel rolls, say, from port to
starboard, the water, having a free surface, rushes in the same direction and acts
against the righting moment operating to return her to the upright position.
The efficiency of the method has been found to depend on the depth of water
in the chamber. This was borne out by observations of the behaviour at
sea of H.M.S. Inflexible^ which had such a chamber, her mean angle of roll
being reduced 20 to 25 per cent, with the chamber about half full, this
being the best result obtained. The value of a water chamber was further
tested in a series of still water rolling experiments with the Edinburgh^ a
warship of the same class as the Inflexible. The chamber in this case was
16 feet long, 7 feet deep, and had a full width of 67 feet; by means of
bulkheads the chamber could also be tried at breadths of 43 feet and
51 \ feet, respectively. Increasing the breadth was found to have a powerful
effect, the extinctive value at 67 feet being three times that at 43 feet. It

THE GYROSCOPE. 267

was also found that the most effective depth of water was that which made
the natural period of the wave traversing the chamber, the same as the
natural rolling period of the ship.

Experiments with a model of the water chamber, fitted on a frame
designed to oscillate at the same period as the ship, showed the efficiency
of the system to be greatest at small angles of inclination.

For various reasons the water chamber method has not become popular.
In the case of warships, changes in design have led to longer natural rolling
periods, and a reduced necessity for special means of extinguishing rolling.
The Inflexible^ for instance, had a G M of 8 feet, while modern battleships
have seldom greater metacentric heights than about 3 feet.

In the case of merchant vessels, the expense of fitting up a water chamber,
and the loss of valuable space which would be entailed by its presence in the
hold, has stood in the way of its general adoption, particularly as the in-
expensive method of fitting bilge-keels has produced satisfactory results.

THE GYROSCOPE. — A proposal for extinguishing rolling motion, which
some authorities think is likely to be widely adopted in the future, has been
brought forward in recent years by Dr. Otto Schlick. In Dr. Schlick's
words, "the method depends in principle on the gyroscopic action of a
flywheel, which is set up in a particular manner on board a steamer, and
made to rotate rapidly. " The principle of the apparatus, and the method of
application, is fully explained by Dr. Schlick in an interesting paper read
before the Institution of Naval Architects in 1904, and to this the reader
is referred for details.

So far the apparatus has, we believe, only been practically applied in
the case of two vessels, but from the reported results of these trials, the
system appears to be a highly efficient one. The first of these vessels is a
German torpedo-boat, 116 feet long, and of about 56 tons displacement.
In this case* the gyroscope, which was fitted for purely experimental purposes,
has a flywheel one metre in diameter, with a proportionate weight to weight
of ship of 1 to 114. It is steam driven, the periphery of the wheel
being provided with rings of blades, and the wheel enclosed in a steam-
tight casing, and worked as a turbine.

The casing containing the wheel is carried on two horizontal trunnions
having their axis athwartships, the steam supply and exhaust passing through
the trunnions as in an oscillating engine. When the vessel is upright and
at rest the spindle of the flywheel is vertical, and the latter when in motion
thus rotates in a horizontal plane ; also the apparatus is free to become
inclined in a fore-and-aft direction. With the gyroscope in action, the effect
of the transverse heeling of the vessel is to cause the apparatus to become
inclined, and moments to be produced reducing the velocity of the vessel's
oscillations and also their magnitude. To control the fore-and-aft move-

*See a paper by Sir Wm. Whyte in the T.I.N.A. for 1907.

268 SHIP CONSTRUCTION AND CALCULATIONS.

ments of the gyroscope and the rotary movement of the flywheel, an
arrangement of brakes is provided.

At the commencement of the experiments the torpedo boat was rolled in
still water. With the gyroscope at rest, a double-roll period of 4*136 seconds
was obtained ; with the apparatus in action, and the flywheel running at
1600 revolutions a minute, the period was found to be 6 seconds, an in-
crease of 45 per cent.

The roll extinguishing effect was found to be enormous. Starting from
an angle of 10 degrees, with the gyroscope at rest, it took twenty single
oscillations to reduce the inclination to half a degree ; with the gyroscope
in action, the same angle was reached in little more than two single oscilla-
tions.

The sea trials were quite as remarkable ; when through the state of the
sea, the vessel was caused to roll considerably, the effect of the action of
the apparatus, when brought into play, was practically to extinguish the
rolling motion. On two occasions, for instance, the vessel, with the gyroscope
fixed, reached arcs of rolling of 30 degrees, which, on the apparatus being
allowed to act, became immediately reduced to 1 degree or ij degrees.
The results of other observations were equally convincing.

The other vessel referred to as being fitted with Dr. Schlick's apparatus
is the coasting passenger steamer LochieL Very few details of the gyro-
scope in this case are available, but it is stated that the flywheel is driven
electrically. From the reports, the roll-extinguishing effect appears to be
quite as great as in the torpedo-boat. On occasions the vessel was found
to be rolling through arcs of 32 degrees, the gyroscope being at rest, and
the effect of bringing it into action was to reduce the arc to from 2 to
4 degrees, oscillations scarcely perceptible to those on board.

It remains to be seen how far this unique system of extinguishing rolling
motions at sea will be adopted in the future, but it is not unlikely that the
success of the Lochiel trial may lead to the installation of gyroscopes in other
vessels of the same class, and also in steamers engaged in cross-channel
passenger traffic.

PITCHING AND HEAVING.— A vessel among waves will have motions
in many directions, depending on her position with regard to the crest lines.
If broadside on, the principal motions will be those of transverse rolling,
but there will be also more or less vertical dipping oscillations due to the
wedges of immersion and emersion being instantaneously dissimilar in volume.
If head on to the waves, while there will be some transverse rolling, the
chief motions will consist of pitching, i.e., longitudinal rolling about a transverse
axis through the centre of gravity, and heaving, due in part to the dipping
motions above mentioned, and in part to the excesses of weight and buoyancy
which occur as the vessel rides over the waves. If, however, the vessel
lie in an oblique direction relatively to the wave crests, simultaneous skew
rolling and pitching motions will be set up, as well as heaving.

The conditions in each of these cases will be modified to a greater or

PITCHING AND HEAVING. 269

less extent by the forward motion due to the propeller. We have seen how
transverse oscillations are thus affected, and we shall consider presently the
influence of speed on pitching and heaving.

The period of unresisted pitching in still water may be determined by
the formula which gives the period of similar transverse oscillations, provided
K be the radius of gyration about a transverse axis through the centre of
gravity, and M the longitudinal metacentric height.

7", as before, being the period in seconds, the formula is —

T = ri4io /— ■ ;
6 J a I

9 M'

As an example, take a vessel of 4000 tons displacement, with a radius
of gyration of no feet, and a longitudinal metacentric height of 285 feet.
In this case —

T , / IIO X IIO , j

/ = V1416 / 7T— = 3'6i seconds.

•* J 32-2 x 285 "*

The corresponding transverse rolling period for this vessel is about 8
seconds, i.e., more than double the other ; and this, in most cases, is the
proportion between the two periods.

In pitching, a vessel always tends to place her masts normal to the
effective wave slope. When a vessel is long in comparison with the waves,
the effective wave slope will depart very little from the horizontal, and the
pitching will be slight; when the opposite is the case, i.e., when the vessel
is short relatively to the waves, the extent of the pitching will be governed
by the natural pitching period, the period of the waves, and the course and
speed of the vessel relatively to the waves. If the wave period be long,
and that of the vessel very short, she will follow the slope of the wave ;
but if the wave period be naturally short, or, if it be made so by the t speed
of the vessel, pitching is likely to become excessive, as the vessel will fail
to rise on each successive wave crest ; her ends will thus become buried,
and the periodical impulses received from the waves will conduce to larger
longitudinal oscillations. Pitching, then, which is in the first instance caused
by the passage of waves, will, like rolling, become excessive if the wave period
keeps time with her natural pitching period.

Every seaman likes his vessel to be lively fore and aft, i.e., to have a
short pitching period. In such a case a vessel will follow approximately the
wave slope, especially if she be short relatively to the wave length, and will
rise on the waves instead of burying her ends into them. This vessel will
be drier than a slower moving boat, and will not be subjected to the same
hammering stress which continual plunging into waves is bound to set up.

In order to obtain a short pitching period, thus seen to be desirable,
the radius of gyration must be reduced or the longitudinal metacentric height
increased. This follows from the formula for the period given above. The

270 SHIP CONSTRUCTION AND CALCULATIONS.

metacentric height cannot be affected to any appreciable extent, as the length
and displacement, on which its value depends, are fixed by more important con-
siderations than those of pitching ; the radius of gyration, however, may obviously
be reduced by concentrating the heavy weights amidships, and in a merchant
ship this should be done as far as possible in stowing the cargo.

Of course, just as in the case of transverse rolling, a master may frequently
help matters. Should pitching become excessive when he is sailing head to
sea, due to synchronism, he may change to an oblique course and thus lengthen
the apparent wave period, and give his vessel time to rise on the waves ; or,
without changing his course, he may reduce speed and attain the same end.

On the subject of vertical heaving and dipping it is unnecessary to say
much. From what we know of synchronism, it is clear that motions of this
kind are likely to be excessive, if the period of dipping is in approximate
unison with that of the waves. Pronounced heaving will not endanger a
vessel's safety, although, as was noticed in a previous chapter, the longitudinal
bending moments, and therefore the stresses brought upon the hull, may be
considerably affected thereby. It may be said that, as usually constructed,
vessels are provided with a sufficient margin of strength to meet all such
demands.

QUESTIONS ON CHAPTER X.

1. How would you set about obtaining the instantaneous axis of an oscillating ship?

2. Write down the formula for the period in seconds of a single oscillation of a ship
j.\ the supposition that there is no resistance. What use is made of this formula by the
naval architect ?

3. Given that the metacentric height in a certain vessel is 2 feet and the radius of
gyration 18, calculate the period of a single roll. Ans. — 7 seconds.

4. The radius of gyration of a vessel is 16, and the single roll period 5 seconds ; find
the metacentric height. If the metacentric height be reduced one foot, what would be the
periodic time? Ans. — 3"i4 feet; 6 - o6 seconds.

5. What is Isochronism? To what inclinations are vessels of ordinary form isochronous?
In rolling through large angles how will the period be affected?

6. Explain briefly the modern theory concerning the form and action of sea waves?

7. Calculate the period in seconds and the speed in feet per second of a wave 500
feet long. Ans. 9S8; 50-6.

8. What is meant by the term "effective wave slope"? Explain why a vessel among
waves tends to place her masts parallel to the normal to the wave slope as virtually her
upright position of equilibrium.

9. Show that the behaviour of a vessel at sea depends largely on the relation between
her still-water period and the period of the waves she may encounter.

10. What difference in the behaviour of a vessel would you expect if her single roll period
were —

(1) longer than the half period of the waves;

(2) much shorter than the half period of the waves?

QUESTIONS. 271

11. Explain the terms "steady," " crank," and "stiff," as applied to a vessel's condition
when among waves at sea.

12. Under what circumstances is the rolling of a ship likely to be most severe?

The single roll period of a vessel is 6 seconds ; how would you expect her to behave
if broadside on to a regular series of waves of 12 seconds period?

13. Referring to the previous question, if the maximum slope angle of the waves be 5
degrees and the vessel upright when a wave reaches her, in what position would you expect
her to be after the passage of six waves, assuming the water to offer no resistance?

14. If a shipmaster finds the rolling of his vessel to become exceptionally severe, to
what cause may he justly attribute it? What steps should he take with a view to reducing
the rolling motion ?

15. Give an analysis of the resistance encountered by a vessel when rolling freely.

16. What are bilge-keels ? In what way do they affect the rolling motions of vessels ?
Discuss the effect of motion ahead on the action of bilge-keels.

17. Bilge-keels are more effective in reducing rolling in small vessels of short period than
in large slow-rolling vessels. Explain why.

18. What are water chambers? Show how they tend to diminish the rolling of ships
in which they are installed.

19. Explain the terms " pitching " and "heaving." Give the formula for the pitching
period of a vessel in seconds. The pitching period of a vessel being 4 seconds, and her
longitudinal metacentric height 400 feet, calculate the radius of gyration.

A n s.— 144*4 feet.

20. Is a short or long pitching period preferable? Give a reason for your answer?

21. Under what circumstances is a vessel likely to pitch excessively? Explain how a
master, who has his vessel well under control, might help matters in such a case and ob-
tain easier fore-and-aft motions.

CHAPTER XL

Loading and Ballasting.

IT should now be clear that to efficiently load a vessel does not mean
simply to fill her with cargo in the shortest possible time. In the

previous chapters we have endeavoured to show that the nature of
a vessel's sea qualities depends upon the manner in which the weights,
including the cargo, are distributed, so that skilful stevedoring is almost as
important as efficient designing.

We have- already seen that the characteristics controlling a vessel's sea
qualities have a conflicting interdependence, which makes it difficult in any
given case to arrange for the values necessary to the best all-round results ;
that with great stability heavy rolling is frequently associated, and with great
steadiness a dangerously small margin of stability. It is thus clear that
considerable care and experience is necessary in order to put cargo properly
into a vessel. The superintendence of this work should, therefore, be en-
trusted only to thoroughly-experienced persons, and owners who take no pre-
cautions of this sort may find the subsequent behaviour of their vessels to
be scarcely all that might be desired.

An intelligent and experienced officer can, with care, usually do much
to bring about a satisfactory condition of his vessel. Even if he does not
gain all he may strive for, his vessel should still be safer and more com-
fortable than if loaded in any haphazard way.

GENERAL CARGOES. — In loading general cargoes, an officer who knows
his business will be guided by the characteristics of his vessel. If she be
narrow and deep, he will place the heavy weights low in the holds and the
lighter weights higher up, thus ensuring a comparatively low position of the
centre of gravity, necessary on account of the metacentre being low in
position in vessels of this type. If the vessel be broad and shallow, the
metacentre will be relatively high, and to obviate a too great value of G M
he will aim at a higher position of centre of gravity, placing the heavy
weights higher in the vessel.

Besides this, following the principles of Chapter VIIL, he will see that
the weights are distributed longitudinally in such a way as to secure a
suitable trim. Thus, with sufficient stability, steadiness among waves and a
satisfactory fore-and-aft flotation may be secured.

The vessel's steadiness may be further improved, without affecting the

272

GENERAL CARGOES. 273

stability, if, without raising them, the heavy items of cargo can be banked
against the ship's sides, as the radius of gyration is thus increased and the
roll period lengthened. Actual experience appears to indicate that, in
ordinary cases, very little can thus be done to improve a vessel's condition,
but the effect of "winging" the weights should not be lost sight of.

The nature of a cargo, it is hardly necessary to point out, is always
a determining factor of the style of loading. It is also admitted that
circumstances may not always be favourable to good stowage. Suitable
cargo may not be available for shipment at the correct time, and, in con-
sequence, the heavy items may occupy positions either too high or too low,
and at the centre of the vessel rather than at the sides ; but such a state
of things may be considered exceptional. When the weights and other
particulars of the various items for shipment are available, a good plan is
for the officer in charge to make a rough estimate of the position of the
centre of gravity. In this way the best places for individual items of cargo
may be determined before commencing operations, and, although in the
process of loading departures may require to be made, these may readily
be allowed for. On completion of the stowage, the metacentric height may,
as previously suggested, be checked by means of an inclining experiment,
and, if necessary, corrected by transposing some of the weights. Also, the
roll period may be ascertained by forcibly heeling the vessel and counting
the number of rolls as already described. It is to be feared the value of
such experiments is not fully appreciated. Owners make much of the
trouble and loss of time involved, and do not give the encouragement they
might to their commanding officers, and hence we find well-proportioned and
designed vessels developing tendencies to excessive rolling, which the exercise
of a little care at the time of loading would have done much to obviate.

It cannot be doubted that the carrying out of the experiments above
described would afford invaluable experience to a commanding officer as to
how particular kinds of cargo should be stowed in his ship to obtain the
best results at sea. Such an officer - might be said to "know his own
ship." It sometimes happens, however, that a man is called upon to take
charge of the loading of a ship of whose qualities he is in total ignorance.

In such a case an officer should be quick to notice changes in the vessel's
condition during the process of loading. If he should observe her to
suddenly list to port or starboard, he may take it her stability, in the
upright position at least, is dangerously small, the sudden movement being
caused by the raising of the centre of gravity above the metacentre, and
the vessel being put into a state of unstable equilibrium. Her stability
curve will resemble fig. 224, that is, she will be unstable from the upright
to the angle at which she has come to rest. The officer must on no
account attempt to cure such a list by moving weights to the high side,
as he might quite correctly do if the list had been a gradual one due to
uneven loading. In the present case the raising of the weights would
make matters worse, and, if the reserve of stability were small, might
S

2 74 SHIP CONSTRUCTION AND CALCULATIONS.

culminate in actual disaster. The only cure is to bring down the centre
of gravity by lowering the position of weights already on board, or shipping
additional weights low down in the holds. A good way is to run up a
compartment of a ballast tank, but, as will be seen later on, this might
be dangerous if the stability reserve were small.

HOMOGENEOUS CARGOES.— In the foregoing remarks we have as-
sumed a more or less general cargo. The case, however, is different with
certain homogeneous cargoes, as we shall now proceed to show.

Suppose, for instance, a vessel has her whole cargo space filled with a
homogeneous cargo, of such density as to just bring her to the load water-
line. This is a trying condition of loading, as an unfavourable position of
the centre of gravity cannot now be corrected by shifting about the cargo.
The only plan open is to discharge part of it, and this few owners would
contemplate with any satisfaction. Such a resort, however, unpleasant though
it be, would, under such circumstances of loading and position of centre of
gravity, be unavoidable if the safety of the ship at sea were to be con-
sidered at all.

Of course, vessels intended frequent] y to load homogeneous cargoes of
this critical density can always be designed to carry a full cargo with perfect
safety. The naval architect would, in such a case, make this the one con-
dition in which the vessel should have sufficient stability and trim properly,
since it is the only one over which stowage has no control.

The importance of good design has been demonstrated by the results of
actual experience. The late Dr. Elgar, in a paper on "Losses at Sea," read
before the Institution of Naval Architects in 1886, made an analysis of
British shipwrecks over a certain period, and showed conclusively that many
of the disasters were due to bad design. Few of the vessels lost were, indeed,
of such proportions as to admit of sufficient stability when fully laden with
a homogeneous cargo such as above described, and many of them were so
laden.

It should be mentioned that the proportionately narrow and deep class
of vessels, to which these mainly belonged, is no longer popular ; the modern
tendency is towards greater breadth, and this is in the right direction.

With homogeneous cargoes of other densities, as with general cargoes,
something may be done to correct a high position of the centre of gravity
due to faulty design. With those of lighter density, for instance, the whole
cargo space may be filled as before, and the margin of draught taken up
by running in water ballast ; if the vessel has no tanks, heavy dry ballast
may be put in the bottom of the holds before the cargo is loaded. With
cargoes of greater density, the whole internal space will not be required, and
so the position of the centre of gravity can be affected by leaving an
empty space in the holds, or in the 'tween decks, according as it is desired
to diminish or increase the value of G M.

SPECIAL HOMOGENEOUS CARGOES— OIL.— Bulk oil, as a freight, is
becoming increasingly important, and special care is necessary in dealing with

OIL CARGOES IN BULK. 275

it. In explanation of this, suppose an oil-carrying vessel, in the process of
loading, to be slightly heeled by some external means. Fig. 232 illustrates
the case and is a section through a partially filled compartment. It will
be noted that the act of heeling has transferred the small wedge of oil
SiOSs across the ship into the position S 2 $& causing G, the centre of
gravity, to be drawn out in the same direction to G v The forces of weight
and buoyancy act through G x and the deflected centre of buoyancy, and, as
drawn, form a couple tending to right the vessel, the arm of the couple
being m Z v

It is thus seen that the effective centre of gravity, so far as the initial
stability is concerned, is raised to m, and the metacentric height reduced
from GM to mM. If we assume that the liquid in the hold is of the
same density as the water in which the vessel is floating, the reduction Gm
in feet may be obtained in any actual case from the formula —

Gm = y,

where 1 is the moment of inertia of the free surface of the fluid in foot units,
and V the volume of displacement of the ship in cubic feet. This formula is

* This formula is obtained as follows: — Referring to fig. 232,

Let 6=hatf breadth of free surface in feet.

/ = length of free surface in feet.

w l = weight per cubic foot of liquid cargo in lbs.

lf„ = weight per cubic foot of water in which vessel is floating in lbs.

V = volume of displacement in cubic feet,

Assuming the vessel to be heeled as in fig. 232, and compartment to be rectangular

at the level of the free liquid surface,

Volume of wedge 8 % 0$ a or S z 0S t = ^b 2 fd cubic foot.

Weight of wedge SJS, or S 2 08 4 = $b*f6 wjbs.

Moment of wedge 8.08, or S..0S, about"* „ , „ , x , ,.

f j ft ■ k In \=\b*iew±b foot lbs.

fore-and-aft axis through U )

And, since weight of vessel = Vxw % lbs.

Let a vertical line be drawn through G lt and call the point in which it intersects the

middle line, m. Then, the inclination being small,

GG^Gmd,

GG, $6 3 / w,
therefore, G m =—j- = -y- x —

But §6 3 / is the moment of inertia of the liquid surface about the axis through 0, i.e., the
middle line. Calling this /', we get by substitution,

Gm= v x--,

which becomes —

when, as assumed above, liquid cargo and water in which the vessel is floating are of same
density.

276

SHIP CONSTRUCTION AND CALCULATIONS.

seen to be similar to that for the height of the transverse metacentre above
the centre of buoyancy, except that G takes the place of B as the point
from which the resulting distance must be measured. Clearly, the greater
the value of /', that is, the larger the free surface of oil, the greater will be
the reduction in the metacentric height. It should be specially noted that
the reduction does not depend on the quantity of oil in the compartment,
as a small quantity having a large free surface will have more effect than a
large quantity with a small free surface.

Practical Example. — A midship compartment of a vessel of 4,500 tons
displacement is partly filled with liquid of the same density as the water in
which the vessel is floating. It being given that the free surface is 30

Fig. 232.

feet long, 38 feet broad,, and rectangular in shape, estimate the reduction
in metacentric height. Applying the formula, we get —

Gm = ^ * * ^-=-87 feet.

12 X4500X35 '

If the liquid in the compartment were oil different in density from the
water supporting the vessel, the above value would require to be multiplied
by the ratio of the density of the oil to that of the water. Thus, if the
cargo were petroleum, and the vessel afloat in salt water, the reduction in
metacentric height would be —

G m = '87 x -8 = -69 feet,

the ratio of the density of petroleum to that of salt water being '8.

OIL CAkGOES IN BULK.

277

In the above case there is assumed to be no middle line bulkhead. Such a
bulkhead, however, is never omitted in modern oil-carrying vessels, as it is of
great value in minimising the detrimental effect of a free surface. This is shown
in fig. 233. The continuous line, S- t S^ indicates the oil surface with the vessel
upright, and the two lines S 3 S^ S 5 Sq } the surface when the vessel is heeled,
the presence of the bulkhead restricting the movement as shown. The wedge
transferred here from one side to the other of each portion of the divided
compartment is half the breadth and one-fourth the volume of that of the
previous case; also the travel of the centre of gravity of the wedge of fluid
is a half, and the moment an eighth. But two wedges of fluid move instead
of one, so that the total moment is one-fourth of what it was in the previous
case. The reduction in metacentric height due to the restricted oil surface,
since it varies directly as the moment, is thus also a fourth.

Fig. 233.

w /

|(U

/ w r

$r^— V- — It

/ sJ-

— k

I**.

From the foregoing considerations, there follow two results of importance.
The first is the advisability of restricting the lengths of oil compartments; the
second, the necessity of exercising great care in loading fluid cargoes. In
conducting the latter operation, it is highly important to keep the vessel up-
right, as a slight inclination caused by the movement of a weight of moderate
amount on board is considerably accentuated by the action of the liquid
cargo, which rushes in the direction of the inclination.

It is customary to draw out a diagram showing the angle to which the
vessel may heel as the liquid rises in the hold. If the vessel has sufficient
stability, it may be possible to fill two holds simultaneously. When the liquid
is first poured into the vessel its free surface is small, and the reduction in
metacentric height, due to loss of moment of inertia of surface, may be less
than the increase due to the fall in the centre of gravity consequent on the

278 SHIP CONSTRUCTION AND CALCULATIONS.

admitted liquid being low in the vessel; but as the liquid rises, its upper
surface broadens rapidly, and m quickly overtakes and passes M, the vessel
becoming unstable.

In a case* investigated by the late Professor Jenkins, Yft coincided with
M when the liquid reached a depth of 15 inches. Heeling then began, and
rapidly increased as the oil rose in the hold, the vessel reaching a maximum
inclination of 19!°. After that she began slowly to right herself, finally re-
turning to the upright when m had passed below M, which took place when
the liquid came within a few inches of the top of the tank.

A point of special importance in loading oil vessels, which, perhaps, need
scarcely be pointed out, is that adjacent compartments should be filled or
emptied simultaneously ; for if one side only were dealt with the inclining
effect would naturally be great. Vessels have frequently been inclined to
dangerous angles when this precaution as to loading has been neglected ; and
there are cases on record even of actual capsizing from this cause. Of course,
when an oil compartment is quite full, no movement is possible, and the oil
becomes virtually a solid homogeneous cargo.

Expansion Trunkways. — A point which must not be overlooked in
connection with bulk oil cargoes, is the loss due to evaporation, and unless
specially provided against, the reduction in bulk may lead to free surfaces
in the holds. Accordingly, every oil compartment has one or more open
trunkways rising above it, and sufficient oil is pumped into the vessel to fill
the holds and partially fill these passages. The horizontal areas of the
trunkways are kept as small as possible, consistent with the volume of oil
in them above the level of the tops of the compartments being fully suffi-
cient to allow for loss due to evaporation without bringing the oil level
below the trunkways. These trunkways, too, being open to the holds, also
serve the purpose of allowing the oil to freely expand and contract in
volume with change of temperature.

GRAIN CARGOES.— Dr. Elgar, in the paper previously referred to,
pointed out that between the years 1881 and 1883, the period covered by his
analysis, vessels carrying grain had a greater number of losses than all other
cargoes except coal. This is striking, as the number of vessels carrying grain
is a small proportion of those engaged in the coal trade, and points to the
existence of special characteristics in the nature of grain cargoes and their
stowage. Investigation has proved these surmises to be correct.

It is found that bulk cargoes, such as grain, even when loaded with
care, have a tendency to settle down during a voyage and to leave empty
spaces immediately under each deck. These spaces have been estimated at
5 to 8 per cent, of the depth of hold, and in fairly large vessels may,
therefore, be of considerable magnitude. After such settlement, the grain has
a free surface, and it is here that the danger lies, for when the vessel is

* See a paper in the Transactions of the Institution of Shipbuilders and Engineers in
Scotland for 1889.

GRAIN CARGOES. 2?rJ>

rolling at sea, the grain tends to put its surface parallel with the wave
slope, and, if the rolling is heavy, shifting is the inevitable result.

The angle to which the free surface of grain must be inclined before
sliding motion will ensue, may be easily obtained. If wheat, for example,
be poured on to a floor until there is a heap, it will be found to take
the form of a cone-shaped pyramid. When sliding has stopped, the angle
which the side of this pyramid makes with the floor, is called the angle
of friction or repose, of this kind of grain ; for, if more wheat be poured
on to the heap, the angle of the cone will be increased, and the particles
will run down the side of the cone until the same angle as before is at-
tained. This is one of the principal differences between a liquid and a
grain cargo. On the slightest inclination of the vessel, liquid puts itself
parallel with the water surface ; with grain the tendency is the same, but
friction between the particles prevents any movement until a certain inclina-
tion is reached; this inclination, in fact, if the vessel be heeled in quiet
water, being the angle of repose of the grain. The value of this angle has
been obtained for various kinds of grain; for wheat it is 23^ degrees, for
two kinds of Indian corn, 26^ degrees and 28-J degrees respectively, for mixed
peas and beans, 27 J degrees.

The late Professor Jenkins, who investigated this subject,* drew attention
to some points of importance with regard to the sliding angle. He showed
that the accelerative forces, which act on a vessel and her cargo when
rolling at sea, qause shifting to take place at a much smaller angle than
the still-water angle of repose. In the case of grain with an angle of repose
of 25 degrees, he found it to be, in a certain vessel of which the par-
ticulars as to stability and radius of gyration were assumed, as low as i6|
degrees. He also found that heaving motions, when accompanied by rolling,
will, at a certain point during each oscillation, cause still further diminu-
tion of this angle. In the example above, it proved to be rather less
than 14^ degrees. As this angle is frequently exceeded by vessels rolling
among waves, the probability of shifting, where there is a free surface,
becomes manifest. It should be mentioned that shifting would take place
in the above vessel at 14 J degrees at one point only during the oscilla-
tion, namely, when she had arrived at the end of a roll and was about to
return ; also, that the whole surface would not slide at this angle, but only
a portion of it at the upper part of the side about to descend. At any
other part, the angle of shifting would be greater, reaching a value in excess
of the still-water angle of repose at the other extreme of the free surface, /,<?.,
on the side about to ascend. Of course, when the vessel became inclined to the
other side of the vertical, this state of things would be reversed. On the
whole, the effect of rolling appears to increase considerably the tendency
to shifting.

* See a paper on the Shifting of Cargoes in the Transactions of the Institution of Naval
Architects for 1887.

2S0 SHIP CONSTRUCTION AND CALCULATIONS.

Piofessor Jenkins showed further that the decrease of angle at which
sliding begins is greater, the greater the stability., but at the same time
pointed out that the effect of a shift of cargo is more serious in the case
of a vessel of small stability than in that of one of great stability. He also
showed that the part of the cargo most subject to movement is that above
the centre of gravity, which, in double-decked vessels, would apply to the
'tween decks. Government Regulations prohibit the carriage of grain in
bulk in 'tween decks except such as may be necessary for feeding the
cargo in the holds and is carried in properly constructed feeders ; generally,
it is largely carried in bags ; dangerous shifting of the cargo at this part
is thus obviated. The stowage of grain in the holds of vessels having a
'tween decks requires special care.* In the case of single-decked vessels,
when shifting of cargo has taken place the effect may be rectified by open-
ing the hatches when the weather permits, and filling up the empty spaces
with bags of grain carried for the purpose. Where there is a 'tween decks
this cannot be done, as the holds are inaccessible, and shifting once begun
cannot be corrected. It is usual to fit trimming hatches through the lower
deck, and these to some extent allow the settlement in the holds to be
made up from the cargo in the 'tween decks, the grain in way of the
trimming hatches being in bulk ; but empty spaces under the beams are
still likely to exist between these hatches.

Allowing for certain exceptions, the Government Regulations require one-
fourth the grain to be carried in bags in all spaces which have no efficient
feeding arrangements. In such cases, before stowing the bags, the grain
must be trimmed level and covered with boards. For the reasons given
above, this rule, which is calculated to prevent serious shifting of cargo,
should obviously apply specially to compartments constituting the lower holds
of vessels having one or more 'tween decks.

As a safeguard against the effects of possible shifting of cargo, grain-
carrying vessels, whether the grain be in bags or in bulk, are required to
have a centre division in the holds and in the 'tween decks, which restricts
the extent of the movement of a grain cargo much as it restricts the
movement of a cargo of liquid. Generally, the centre division consists ot
portable wood boards fitted edge on edge and reeved between the centre
line of pillars, which are reeled for the purpose • but in some modern
vessels it consists of a permanent steel bulkhead (see page 161) except in
way of the main hatchways, the exigencies of stowage demanding portable
boards at these places.

COAL CARGOES. — Owing to its density, coal can, in general, be
stowed at such a rate as to ensure a certain amount of empty space in the
'tween decks, the vessel at the same time being down to her load mark.
There is, therefore, no apparent reason why coal-laden vessels should not

* See a paper by the late Mr. Martell in the Transactions of the Institution of Naval

Architects for 1SS0.

TIMBER CARGOES. 28 1

have sufficient stability. From the great number of such vessels which have
been lost, some of them known to possess a fair amount of stability at
the start of their fatal voyages, it has been conjectured that shifting of the
cargo may have been the cause of not a few of the disasters. Colliers
are not usually fitted with shifting boards, and there is no restriction placed
on the stowage of coal in the 'tween decks, as with grain ; and, although
the angle of repose of coal is considerably greater than that of grain, the
diminishing process it undergoes during rolling motions at sea may doubt-
less often bring it within the range of a vessel's oscillations in stormy
weather. From which considerations it would appear that the suggested
reason of shifting for the loss of many coal-laden vessels may be quite near
the mark.

The lessons to be deduced here by the ship's officer are, first, to aim
at so loading his vessel as to ensure easy motion when among waves at
sea; and second, to see that no vacant spaces are left under the decks,
as these inevitably lead to shifting of the cargo.

TIMBER CARGOES.— In the case of cargoes of the heaviest woods,
the full hold capacity is not required, and loading should simply follow the
lines already indicated for ordinary heavy deadweight cargoes. With cargoes
of mixed timbers, satisfactory conditions of stability and trim can always
be attained by a proper distribution of the light and heavy woods. In the
case of cargoes of the lightest timbers, however, the problem of stowage
becomes more difficult, because, as well as full holds, there is usually a
considerable deck load carried.

Many vessels that are good deadweight carriers, and quite suitable
for general trades, could not, without a considerable amount of ballast, be
safely employed to carry a cargo of timber of the last description, the
high position of the centre of gravity, due to the presence of the deck
cargo, making the stability quite inadequate. In some cases, indeed, there
might be actual instability in the upright position, and no seaman would
care to face a sea voyage in a ship having a pronounced list to port or
starboard. In this trade, vessels should be specially broad in relation to
draught, as this ensures a relatively high position of metacentre, and a
sufficient margin of stability, without having to resort to ballast. It should,
of course, be noted that a deck cargo of wood, when well packed and
securely lashed in place, affords valuable surplus buoyancy, which has a
marked effect on the form of the stability curve, giving it increased area
and range, although at initial angles, owing to a high position of the centre
of gravity, the righting arms may be small.

We have already referred to curves of this type (see curve No. 3, fig.
223), and have pointed out that in such cases the value of GM may, with
perfect safety, be quite small; and as this ensures a long rolling period, a
vessel so circumstanced should prove an easy roller, and thus a comfortable
boat in a seaway

The value of G M in a case like this, should be limited by the con-

282

SHIP CONSTRUCTION AND CALCULATIONS.

sicleration of the vessel being stable, and not too tender, in the upright
position throughout the voyage. That is, over and above a sufficient margin
to cover diminutions from causes that may be anticipated, such as the
burning out of the bunker coal and the increase in weight of the deck
cargo through becoming saturated with sea water, a certain minimum value
of G M, such as previous experience with the vessel may suggest, should be
provided. A metacentric height of greater value is unnecessary, and, to the
extent of the excess, might be considered as actually detrimental to the
vessel.

EFFECT ON STABILITY OF A SHIFT OF CARGO.— To calculate
the quantity involved in any particular shift of cargo is not always an
easy matter. In the case of oil cargoes, where there is a free surface, the

Fig. 234.

quantity shifted through the heeling of the vessel may be accurately deter-
mined, since the oil cargo is always horizontal, but, as we have seen,
owing to friction between the parts, dry homogeneous cargoes do not move so
quickly nor so definitely as oil, and the same rules cannot be applied.
Still, with grain, an approximation may be made to the heeling effect of
the worst shift of cargo that is likely to take place in a given case, when
the plans of the vessel are available.

Fig. 234 is the midship section of a grain-laden vessel. The horizontal
dotted line a Ct r shows the grain level, assuming the settlement to have
taken place evenly, and is drawn at a distance below the deck, correspond-
ing to the anticipated maximum settlement of the grain, i.e., about 5 per
cent, of the depth of the hold. b b x is the ultimate line of the grain
surface when the cargo has shifted, and may be taken to lie to the
horizontal at the angle of repose of the grain. The wedge of grain a x (Ibx

EFFECT OF A SHIFT OF CARGO. 283

now occupies the position ctcbd, its centre of gravity moving from #, to
</ 2) and the common centre of gravity of vessel and cargo moving from
G to G v

If w = weight of cargo shifted in tons,

W ~ displacement of vessel in tons,
9i3% = travel of centre of gravity of shifted cargo in feet,

G G x will be parallel to the line joining g x and g 2 ; the point G will thus
be raised relatively to the keel as well as moved laterally. For small angles,
however,

GG X = GM x e (nearly).

From which equation Q } the angle at which the vessel comes to rest, may
be obtained.

This lateral movement of the centre of gravity obviously means a
reduction of the righting arms, and in a vessel originally tender might en-
danger her safety. In the case assumed there is no middle line bulkhead;
if such were fitted, the angle of heel, as in the case of a liquid cargo,
would be reduced.

Practical Example. — In a grain-laden vessel of 48 feet beam, 9000 tons
displacement, and 18 inches G M, 50 tons of cargo is shifted transversely
through a distance of 27 feet. Calculate the angle of heel.

In this case —

GG 5^H7 d GM x Q
9000

KO X 27

.*. « -^— — — « -io
9000 x 1 '5

= 6° nearly.

If the shift of cargo were considerable, it might be necessary to allow for the
vertical as well as the transverse movement in determining the angle of heel.

Let h — the vertical distance between g x and g 2i

d = the horizontal distance between g x and g 2 ;

,r • , r r IV X H r

then, Vertical movement of centre 01 gravity — — 7^ — feet,

TT • r r . w x a r

Horizontal movement of centre 01 gravity == — rr. — feet.

This determines the position of the centre of gravity after shifting has taken

place. Join this point with M, the metacentre, then the angle between this

line and the middle line is the angle of heel required, provided it does not
exceed io to 15 degrees.

to h
wd

*W-

w h

150 >

: 20

2S4 SHIP CONSTRUCTION AND CALCULATIONS.

In the previous example, if the cargo shifted had been 150 tons, the
vertical movement 5 feet, and the transverse movement 20 feet —

tod

Tan Q=-

Substituting values —

Tan a =

i"5 x 9000 - 150 x 5

= ' 2 35

= 1 3 nearly.

Since shifting of cargo may be more or less expected in grain-laden ships,
the stability of such vessels should be carefully considered before setting out
on a voyage, the effect of the shifting of the greatest quantity of cargo
that may be anticipated, estimated, and a safe margin of stability provided.

EFFECT OF BURNING OUT BUNKER COAL.— This is a point of
importance in fixing upon a value of GM with which to start a voyage.
While it is desirable, in the interests of steadiness at sea, to avoid ex-
cessive initial stability, if it is known that consumption of the coal will
entail a diminution of the metacentric height, the latter to begin with should
be sufficiently great to allow for such loss. In many cargo vessels the
centre of gravity of the bunkers is higher in position than the common
centre of gravity of ship and load, and the removal of the coal thus leads
to an increase of the metacentric height and righting levers ; but in some
steamers there are large reserve bunkers extending no higher than the lower
deck, and the rise in the vessel's centre of gravity, due to the burning of
the coal, causes considerable reduction in the metacentric height, reductions
amounting to as much as ih feet being not unknown.* The need of
investigating this question of the bunkers is thus manifest.

Curves showing the condition with bunker coal in and out are now
supplied by many shipbuilding firms with new vessels for the guidance of
the officers.

BALLASTING is the name given to the loading of deadweight other
than cargo, to enable a vessel to make a voyage in safety.

Unfortunately, vessels do not always find an available freight at the port
of discharge, and have consequently often to make the return journey, or
proceed, at least, to another port, without a remunerative cargo. Every
seaman knows that it would be imprudent to attempt such a voyage, if it
meant crossing the sea with a chance of rough weather, with an absolutely
" light " ship. He knows, in the case of a sailing-ship, that the stability
would probably be insufficient to withstand the heeling effect of a spread of

* See a paper by the late Dr. Elgar in the T.I.N. A. for 1SS4.

BALLASTING. 285

sails ; in the case of many steamers, that the stability would be dangerously
small, owing to a relatively high position of the centre of gravity ; and, in
all such cases, even assuming sufficient stability, that their slight grip of the
water, and their greatly exposed surfaces, would cause them to be the sport
of wind and waves, with a probability of serious damage before the end of
the voyage. Accordingly, if he cannot get cargo, he takes ballast aboard, i.e.,
sand, gravel, rubbish, or water. Frequently, a combination of water and one
of the other forms of ballast is used. In steamers, bunker coal is useful
in this way.

As to the total amount of ballast required, it should be sufficient to
secure the following : —

(1) An adequate immersion of the propeller in steamers to prevent
racing of the engines and breaking of the tail shaft, and undue
strains being brought upon the stern frame.

(2) A stability curve of suitable area and range, with considerable
righting moments at large angles of inclination in all vessels.

(3) A good floating depth to give grip of the water, and to reduce,
as far as possible, when among waves, the effective wave-slope angle,
which, as we have seen, directly affects the magnitude of the arcs
through which a vessel will roll.

It would appear that a great variety of opinion exists as to the ratio
which should hold between the amount of ballast and the full deadweight. In
a paper by Mr. Thearle on "Ballasting of Steamers for Atlantic Voyages''
in the Transactions of the Institution of Naval Architects for 1903, the
actual figures in 41 cases are given, and show the ballast draught to vary
from "5, to 72 of the load draught, and the amount of ballast from '24 to
•51 of the full deadweight, shelter-deck vessels with high sides having the
greatest proportion of ballast and draught, and small vessels with flush decks
or with very short erections the smallest Mr. Thearle points out, that for
good results, the amount of ballast in ordinary tramp steamers when making
voyages across the Atlantic in winter should not be less than one-third of
the full deadweight, with the vessels 4 to 5 feet by the stern, and the pro-
pellers about two-thirds immersed, experience having shown that damage in
the form of loose rivets at the ends is likely to result where there is a
less proportion of ballast. A point in ballasting not less important than
having a sufficiency of deadweight, is that the latter should be carefully
stowed. The same principles should, in fact, be followed here as in ordinary
loading operations ; that is, the ballast should not be placed too high, or
the stability may be endangered, nor too low, or there may be an undue
depression of the ship's centre of gravity, and a consequent abnormal in-
crease in the metacentric height, a state of things, as we have seen, inevitably
leading to excessive rolling and great straining of the hull.

In vessels having 'tween decks, part of the ballast, if of sand or
rubbish, should be placed there ; in single-decked vessels some special
arrangement should be made to raise the centre of gravity, either by fitting

286 SHIP CONSTRUCTION AND CALCULATIONS.

temporary bulkheads to bank up the ballast in the holds, or by carrying
part of it properly secured on deck, or by any plan which experience and
the special circumstances may suggest. Unfortunately, such precautions are
not always taken, and thus many vessels in ballast are unduly stiff. Nowadays,
particularly in steamers, water ballast is largely used. It has the obvious
advantage over sand or rubbish of being more easily, quickly, and cheaply
loaded and discharged. Moat modern cargo steamers have double bottoms
and peak tanks ; some large vessels have also one or more deep tanks ex-
tending from the bottom of the vessel to the first or second deck; while
in a few instances ballast tanks have been built into the corners under the
deck, also on top of the deck between the hatches, and in other places.
For details of the construction of ballast tanks, the reader is referred to
the chapter on practical details.

A double bottom, of course, except in the special case in which it
extends up the ship's side, is not the best place for ballast. The amount
carried in a double bottom, however, is not of itself sufficient for a sea voyage,
and if the remainder is loaded in deep tanks, or in corner tanks under the
deck, of which the capacity has been carefully considered, a satisfactory
immersion and a metacentric height such as to ensure good behaviour at
sea may be attained.

Where the ballast supplementary to that in the double bottom consists
of stones or rubbish, it should be disposed so as to obtain a suitable
position of the centre of gravity.

Scarcely less important than the vertical distribution of ballast, is the
placing of it longitudinally. In steamers, as already noted, there should be
a preponderance aft to properly immerse the propeller. But this allowed
for, the remainder should be disposed so as to obtain a suitable pitching
period. This period is, we know, lengthened by winging out the weights
towards the extremities and shortened by concentrating them amidships. To
obtain, therefore, a satisfactory quick fore-and-aft motion, and avoid the
constant tendency which a slow-moving vessel has to bury her extremities
in the waves, supplementary ballast, whether water in deep tanks, or stones
and rubbish, should be placed towards amidships, while peak tanks, where
such are required, should be kept within moderate limits.

Unfortunately, concentrating the ballast amidships in this way is likely
to lead to the development of considerable bending moments, but these
cannot well be avoided, and the strength of vessels should be made sufficient
to meet all such demands.

DANGER OF FILLING BALLAST TANKS AT SEA.— Mention has
already been made of this point, which should be abundantly clear from the
remarks on the loading of liquid cargoes. It is to be feared that many
officers do not fully appreciate the danger of this practice. The ballast is
loaded in order to increase the metacentric height and, therefore, the stability,
but, as we have seen, the presence of the free surface during the process
may deprive a vessel of her effective metacentric height and cause her to

DANGER OF FILLING BALLAST TANKS AT SEA. 287

heel to a dangerous angle, if not to capsize. It is important to remember
that the formula —

Gm = L i

as previously remarked, shows that it is the extent of the area of the free
surface, not the magnitude of the quantity of liquid in the tank, which in-
fluences the metacentric height. And commanding officers should see that
when the ballast is out, the tanks are quite empty, particularly in the case
of midship compartments which are of considerable breadth.

As a concrete example, take the following :— In a certain vessel of 7000
tons displacement, it is intended to run up a 'midship compartment of the
double bottom, which is 80 feet long, 35 feet broad, 4 feet deep, approxi-
mately rectangular in shape, and has a capacity for 320 tons of salt water.

Given that the distance between the ship's centre of gravity and the top
of the tank is 12 feet, calculate the reduction in metacentric height when
the water in the tank is 1 foot deep, the metacentre being assumed to
remain at the same height above the base throughout.

We have here to consider two things, viz., the fall in the centre of
gravity due to the admission of the water, and the virtual rise in the
centre of gravity due to the free liquid surface. Taking the centre of
gravity of the admitted water to be at half its depth, we have —

80 x 15*5
Fall m centre of gravity of vessel = — — « = "17 feet.

In finding the effect of the free surface, if we suppose the fore-and-aft
girders, including the middle one, to be pierced with holes, the whole breadth
of the tank will be available in estimating the moment due to the shifting
wedges of water, and, therefore —

80 x -ik x ?.< x -ik n n

' = I2 = 285833 (fo0t units) '

and

. . , - 2^5833

Virtual rise in centre of gravity = —5 = i*i«j feet.

& J 7080 x 35 J

We thus get,

Reduction of metacentric height = 1*15 - '17.

= -98 feet.

This reduction is serious, and in the case of many vessels would cause
instability in the upright position. It is the general practice, however, to fit
the centre line division without perforations. In that case, the virtual rise
in the centre ot gravity would be a fourth of the above amount, or *2Q
feet, and the reduction ot the metacentric height would only be —

•29 - "17 = "12 feet,

showing the powerful effect of a watertight centre division in a double
bottom.

25S SHIP CONSTRUCTION AND CALCULATIONS.

STABILITY INFORMATION FOR COMMANDING OFFICERS. — A

common plan with many shipbuilders in supplying stability information to
new vessels for the guidance of the officers, is to provide diagrams of curves
depicting the nature of the stability under certain anticipated conditions of
loading and ballasting, along with such remarks as may be necessary for
the proper interpretation of the curves. In closing the present chapter,
we shall give two examples of such stability diagrams, and shall discuss
briefly how they may be employed in the actual working of vessels. Fig.
235 is a diagram for a modern cargo steamer of the following dimensions: —
Length 395 feet 6 inches, breadth 51 feet 6 inches, moulded depth 29 feet
3 inches, mean load-draught 23 feet n£ inches. The vessel has a short
poop, bridge, and forecastle, disconnected, and a main 'tween decks ; she is
adapted to carry water ballast in a double bottom and in both peaks.

' Fig. 236 is a similar diagram for a smaller cargo steamer also of
modern design. The dimensions are : — Length 351 feet o inches, breadth
49 feet 3 inches, moulded depth 28 feet 5 inches, mean load-draught 23 feet
6| inches. The erections consist of poop, bridge, and forecastle; there is
also a main 'tween decks, and accommodation for water ballast in a double
bottom and in the after peak.

Curves A to H in each diagram refer to the following conditions : —
1st condition (curve A). — Light ship, />., vessel complete, water in boilers,
but no cargo, bunker coal, stores or fresh water aboard, and all
ballast tanks empty.
2nd condition (curve B). — Same as 1st, but with bunker coal, stores,

and fresh water aboard.
3rd condition (curve C). — Vessel ready for sea, water in boilers, bunker
coal, stores, and fresh water aboard, and the holds and 'tween decks
filled with a homogeneous cargo of such density as just to bring
the vessel to her legal summer load-line.
4th condition (curve D). — Same as 3rd, but with bunker coal, stores
and fresh water consumed, approximating to the condition at the
end of a voyage.
5 th condition (curve E). — Vessel ready for sea, water in boilers, bunker

coal, stores and fresh water aboard, and all ballast tanks • filled.
6th condition (curve F). — Same as 5th, but with bunker coal, stores

and fresh water consumed.
7th condition (curve G). — Same as 3rd, but laden with a coal cargo,

part of the bridge 'tween decks being empty.
Sth condition (curve H). — Same as yth, but with bunker coal, stores

and fresh water consumed.
In the case of the larger vessel, it will be observed that the 3rd con-
dition (curve <?, fig. 235) is a critical one, the stability reserve being very
small. When loading a cargo of the given density, it would probably be
considered desirable, in the interests of the vessel's safety, to remove some
of the cargo from the bridge 'tween decks, and run up a compartment of

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290

STABILITY INFORMATION. 2C)t

the double bottom so as to bring down the centre of gravity and improve
the stability.

In vessels of this size and description, safety demands that the righting
arms at inclinations of 30 degrees and 45 degrees should not be less than
about *8 of a foot. Let us find what fall in the centre of gravity would
be necessary to secure this in the present case. At 30 degrees the righting
lever is '26 feet ; it has thus to be increased by ('8 - "26) = '54 feet.
Assuming the draught to remain unchanged-
Increase of righting arm at» _ .. . r . . ,
. ,. ° °. [ = Fall in centre of gravity x sin. 30 degrees,
inclination 01 30 degrees )

•54 = Fall in centre of gravity x *5,

. . Fall in centre of gravity ~ — = 1*08 feet.

The curve of stability under the new conditions may now be obtained by
increasing the ordinates of curve G throughout by the amount —

1 '08 x sine of angle of inclination.

In this way curve /C, fig. 236, has been derived.

With a stability diagram like fig. 235 or fig. 236 ready to hand, an
officer should be able in most cases to satisfy himself as to the state of his
vessel. In making deductions, however, he must be careful to note that he
can only deal with draughts for which he has curves ; also, that differences
in stowage may quite alter the nature of the stability.

This would appear to limit the utility of the curves, but it may be pointed
out, with regard to draughts, that a ship is usually either light, fully loaded,
or in ballast, so that in this respect the curves should be found generally
applicable. Differences in stowage give rise to more trouble. The stowage
of a general cargo, or of homogeneous cargoes other than those for which
special curves are provided, lead to variations in the positions of the centre
•of gravity from those of the standard conditions at the same draught. To
obtain a stability curve for any such new condition, the amount of the rise
or fall of the centre of gravity from the position of a standard case must
be known, and, with the information usually given, the only way of obtaining
this would be by means of a special heeling experiment. If, however, the
position of the centre of gravity above the base corresponding to the various
conditions were stated, as in the examples given, a change in the position
of the centre of gravity might be approximated to by a simple moment
calculation. It would only remain, then, to deduce the righting levers of the
new curve from those of the appropriate standard curve, by deducting or
adding at each inclination the amount —

Rise or fall of centre of gravity x sine of angle of inclination.

The tables of conditions supplied with the diagrams of stability curves are
very useful in working out problems like the foregoing. Suppose, for instance,
it were intended to load the smaller of the two given vessels with a full

292

SHIP CONSTRUCTION AND CALCULATIONS.

general cargo. It would be first necessary, by means of the capacity plan,
which is part of the equipment of all modern vessels, to approximate to the
positions of the various weights forming the cargo ; then, by a moment cal-
culation, to combine these weights and heights with those of the light ship
(taken from the table) to obtain the height of the common centre of gravity
of vessel and cargo. Such a case is worked out in detail on page 192.

Suppose this done in the present case,* and the centre of gravity found
to be '5 feet above the position corresponding to a full coal cargo (curve
G, fig. 236). The levers of the required curve are equal to those of curve
G, decreased throughout by the amount —

■5 x sine of angle of inclination,
as already described. In the subjoined table the righting levers at 15 degrees,
30 degrees, etc., are tabulated, and the stability curve, marked K, is plotted
in fig 236,

Angles of

Inclination.

De.uTee^

x 5

3°
45
60

75
90

Ordinate* of

Else of C.G.

Ord mates of

Standard C'un e

x Sine of Angle

required

(Coal Oariro).

of Inclination.

Stability Curve.

Feet.

1-VeL

Feet.

■52

' l 3

"39

T'lO

■85

r 7 8

•35

x '43

2"27

"43

1-84

I '40

• 4 3

•92

•70

*5°

As another example, and in this case referring to the larger vessel,
suppose 200 tons of coal to be put into the bridge 'tween decks in excess
of the amount allowed for in curve G (fig. 235), the same quantity being
omitted from the lower hold to keep the mean draught as before. Let the
height through which the centre of gravity of the coal has been raised be
20 feet, then —

Rise of centre of gravity"! 200 x 20

of vessel

10827

= *37 feet.

The levers of curve G (fig. 235), reduced by '37 x sine of angle of inclina-
tion, furnish the data for constructing the stability curve of the new condition
(this curve is not shown in the diagram).

When a diagram of metacentres is available, along with a table of con-
ditions, the GM for any condition of loading and ballasting can be quickly
determined. Thus, if it were decided to load supplementary ballast in the
larger vessel when in the 5th condition, with a view to increasing her grip
of the water and making her more navigable, the new GM might be arrived
at in the following manner : —

*No reference has been made above to the question of trim, but, of course, in all loading
problems this must be kept in view and the cargo distributed to obtain the best results,
approximate calculations being made where necessary.

STABILITY INFORMATION.

293

Referring to the table of conditions, the metacentric height in the given
standard case is found to be 7*57 feet. As this is excessive, the supple-
mentary ballast should be stowed high, particularly as the stability curve (see
E, fig. 235) is of great area and range.

With regard to the amount of the additional ballast, let it be sufficient
to make the total equal to about a third of the full deadweight, this having
been shown to be a good average. Including bunker coal, stores and fresh
water, the deadweight is 7677 tons, the total ballast should therefore be—

7677

" = 2559 tons.

Including bunker coal and water ballast, 1954 tons is already loaded (see
Table), therefore —

Supplementary ballast = 2559 - 1954
= 605 tons,
or, in round figures, say 600 tons.

In the present case, if such would also suit the trim, it would be an
advantage to put the whole amount into the bridge 'tween decks. If only
half of it can be so placed, and the remainder is passed into the main
'tween decks, the height of the centre of gravity will be as obtained below,
the centres of the supplementary ballast being assumed taken from the
capacity plan as before.

Items.

Weights.

Heights of C.G.

above base.

Moments
about base.

Ballast Condition

(from table), -

5 io 4

16-6

84726

Extra ballast in main

'tween decks,

300

26*0

780O

Extra ballast in bridge

'tween decks,

300

33'5

IOO50

5704

102576

102 ^76

Height of centre of gravity above base — — — = 18 feet.
6 . . 57°4

From the deadweight scale, the increased load is found to sink the

vessel 14J inches, which, added to 12 feet of inches, the mean ballast draught

in the table, gives 13 feet 3 inches for the new condition. At this draught

the metacentre is 23*3 feet above the base, so that —

New G M - 233 - 18-0
= 5*3 feet.
There is thus a reduction of over 2 feet in the metacentric height, which,
with the increased immersion, should ensure an improvement in the vessel's
behaviour at sea.

In the same way the effect of loading supplementary ballast in the
smaller vessel may be determined.

As a final example — suppose the smaller vessel, laden with coal, has to

294

SHIP CONSTRUCTION AND CALCULATIONS.

discharge part of her cargo at a certain port, and afterwards proceed to sea
under the reduced load. The question is, how should the unloading be done
so as to leave her in a favourable condition for prosecuting the voyage?

Let the amount to be discharged be 2000 tons. From the deadweight
scale, assuming the vessel to rise evenly, the reduction in draught due to
unloading the coal is found to be 4 feet 9 inches. In order to keep the
draught aft right, the coal should be taken out forward, say 1000 tons from
the fore main-hold, and the remainder from the main and bridge 'tween
decks.

Taking the centres from the capacity plan, and the figures for the
loaded condition from the table, the calculation for the height of the centre
of gravity is as follows : —

Items.

Weights.

Heights of C.G.
above base.

Moments
about base.

Displacement

Cargo removed from

lower hold
Cargo removed from

main 'tween decks
Cargo removed from

bridge 'tween decks

9068

- IOOO

- 600

- 400

18-OI

13-00
25*00

32-00

I633I5

- I30OO

- 15000

- 12800

7068

122515

122515
Height of centre of gravity above base = ~TTZo~ — x 7'33 feet

The new mean draught is thus 23' 6f" — 4' 9", or 18' 9f", and (from
the metacentre diagram) the corresponding height of metacentre above the
base is 19*9 feet.

•. New G M = 19*9 - 17*33
— 2 '57 feet.

This is a larger metacentric height than in the fully loaded condition, but
as the displacement is less, it may be considered satisfactory.

If considered necessary, the trim might also be approximated in the
above case by measuring from the capacity plan the distances between the
centre of gravity of the load waterplane and the centres of the loads to be
discharged, calculating the trimming moment, and dividing it by the moment
to alter trim estimated in any of the ways described in Chapter VIII.

QUESTIONS ON CHAPTER XI.

1, In superintending the loading of his vessel, enumerate the points which should be
kept in view by the commanding officer.

Explain why great initial stability in a vessel is conducive to bad behaviour at sea.

2. How should the items of a general cargo be stowed to obtain the best results at sea?

(1) In the case of a vessel proportionately broad and shallow.

(2) In the case of a vessel proportionately narrow and deep.

QUESTIONS. 295

3. If, in the process of loading, a vessel is observed to suddenly list to port or
starboard, what may be inferred as the probable cause, and how should the subsequent loading
be conducted so as to bring the vessel back to Ihe upright?

4. Whether does a. general or a homogeneous cargo afford greater facilities for loading so
as to produce a comfortable vessel at sea? Give reasons for your answer.

5. A vessel b33 been loaded to her maximum draught, with a homogeneous cargo which
entirely fills her, and the master desires to ascertain the metacentric height before sailing.
Explain how he may readily obtain this knowledge.

If the metacentric height were found to be deficient, what steps should the master take
to correct it?

6. Write down the formula for the reduction in the metacentric height due to the
presence of a free liquid suiiace in the hold.

One of the compartments of an oil steamer is partially filled with petroleum. Calculate
the reduction in the metacentric height due to the free surface, given that the compartment is
situated amidships, is 30 feet long, 42 feet broad, and approximately rectangular in shape at
the level of the oil, and that the total displacement is 6500 tons. Ans.— '65 feet.

7. Show that the presence of a middle-line bulkhead greatly modifies the effect of a tree
liquid surface in the holds. Assuming a middle-line bulkhead in the vessel of the previous
question, what would be the reduction in metacentric height?

8. Enumerate the precautions which should be taken in loading a vessel with oil in bulk.
Why are trunkways fitted in oil vessels ?

9. Explain why it is that grain cargoes loaded in bulk are frequently found to shift
during n voyage when bad weather has been encountered.

A grain-laden vessel of 7000 tons displacement has a metacentric height of 2 feet 6 inches.
If 100 tons of cargo shift transversely through a distance of 18 feet, what will be the angle
of heel, assuming the vessel to have been upright before the shifting took place?

Ans. — 6°, nearly.

10. Show that the burning out of bunker coal may have an important influence on a
vessel's condition.

If the coal in h particular vessel, whose margin of stability is small, is contained in 'tween
decks as well as lower bunkers, how should it be worked out in the interest of the safety of
the vessel ?

11. A steamer 470 feet in length, 15,600 tons displacement, drawing 27 feet 6 inches
forward and aft, has a reserve bunker containing 500 tons of coal. The centre of gravity of
the latter is 10 feet below that of the vessel, and 35 feet before the centre of gravity of the
load-waterplane. The tons per inch is 60, the longitudinal metacentric height is equal to the
length of the vessel, and the transverse metacentric height at the start of the voyage is 1 - 5 feet.
Assuming the transverse metacentre to remain at the same point while the vessel rises to the
lighter draught, estimate approximately the draught and transverse metacentric height when the
coal in the reserve bunker is consumed.

[ ( Forward, 26 feet, 2| inches.

Am,.- { DraUghtS \Aft, 27 feet, 4 f inches.

I Metacentric height, 117 feet.

12. What is meant by ballasting? What should be aimed at in ballasting a steamer for
a sea voyage? How would you expect a vessel to behave if laden with heavy ballast placed
low down in the holds?

13. A cargo steamer of 7000 tons deadweight is to be ballasted for an Atlantic voyage.
With bunker coal, stores and fresh water aboard, and water in boilers, the displacement is
3350 tons, the draught forward 8 feet and aft 10 feet 9 inches, and the centre of gravity
195 feet above the base. Water ballast is then loaded as follows: — 1000 tons in double

2g6 SHIP CONSTRUCTION AND CALCULATIONS.

bottom, centre 2 feet above base and 2 feet before 'midships ; 650 tons in a deep tank abaft
engine-room, centre 14 feet above base, and 54 feet abaft 'midships; 140 Lons in the fore-peak,
centre 16 feet above base, and 163 feet before 'midships. Assuming the centre of gravity of
the parallel layer to be 3 feet forward of 'midships, the average tons per inch in way of in-
creased immersion to be 34, the average moment to alter trim 1 inch 670 foot tons, and the
height of the transverse metacentre above the base with the ballast aboard 19 feet ; estimate
approximately the draught and metacentric height when in ballast trim.

f C Forward, II feet, 4^ inches.

Am.-) Drau g hts (Aft, 16 feet, ij inches.

I Metacentric height, 37 feet.

14. If, while on her voyage, a vessel should exhibit signs of <£ tenderness," show that it
would be unsafe to attempt to remedy matters by running up a compartment of the double
bottom.

Referring to example 11, it is proposed to fill a compartment of the double bottom, with

a view to making good the reduction in metacentric height due to the consumption of the

coal. The tank chosen for the purpose is approximately rectangular in shape, and has a

perforated centre division ; it is 60 feet long, 44 feet broad, and 4 feet deep, and has

capacity for 300 tons of salt water. Assuming the top of the tank to be 18 feet below the

centre of gravity of the vessel, estimate approximately the metacentric height when the tank

is half full, and also when it is full. ( '58 feet.

Ans. — [ , , H ~ c .
\i *p feet.

15. A vessel whose load displacement is 7500 tons is being loaded in dock for a. summer
voyage. If the water in the dock weighs 63 lbs. per cubic foot, to what extent may the
centre of the disc — that is, the legal summer load-line in salt water — be immersed? The
area of the load waterplane is 12,000 square feet.

Ans. — 4 inches.

16. In loading a general cargo, how should the heavy items be disposed longitudinally to
ensure a vessel behaving well at sea? Give reasons.

17. What stability information should be supplied with a new vessel for the guidance 01
the officers?

APPENDIX A.

CHANGE OF DRAUGHT IN PASSING FROM FRESH TO SALT

WATER. — Taking salt water to weigh 64 lbs. to a cubic foot, and fresh
water 62*5 lbs., the number of cubic feet to a ton in each case is —

r, 1 2240

Salt water, -p— - 35,

_ . 2240

Fresh water, j— = 35*84.

Let now W = a vessel's displacement in tons,
then immersed volume in salt water = 35 x W cubic feet,
and immersed volume in fresh water = 35*84 x W cubic feet.
In passing from fresh to salt water, the vessel thus rises out of the water
to the extent —

35-84 W - 35 W = '84 W cubic feet.
If A be the area of the waterplane in square feet, and d the distance
through which the vessel rises in inches —

■84 Wx
A

Let

the

i tons per

inch

of immersion

salt

water ;

then,

r =

T2"
35

420*

A =

420 T,

Substitute

(1).

and we get-

d -

■84 W x 12

420 T ' 41*7 T

If instead of fresh water, river water weighing 63 lbs. per cubic foot be
assumed, the immersed volume will be —

W x ^~ = 35 - W cubic feet,

and the difference in volume in river and in sea water,

- W cubic feet.
9

Substituting this in the expression for rf, we get —

63 T

Thus, if a vessel of 9500 tons displacement, whose tons per inch at the

297

2 9 8

SHIP CONSTRUCTION AND CALCULATIONS.

load draught is 3$, were to pass from river water at 6$ lbs. per cubic foot
into sea water, she would rise —

9500

4*3 inches.

63 x 35

MEAN DRAUGHT. — In reading a displacement from a displacement
scale, it is the usual custom to employ the vessel's mean draught, i.e., the
sum of the actual draughts read off at the stem and stern divided by 2.
This assumes the waterplane drawn at this mean draught to cut off wedges
of equal volumes forward and aft, which in ordinary cases would not happen.
To cut off a displacement closely approximating to that of the vessel when
out of the normal trim, a line should be drawn parallel to the base through
the point in which the actual waterline intersects the locus of the centres
of gravity of waterplanes.

Fig. 237 shows a vessel trimming by the stern ; W L is the line of
flotation at which it is required to know the displacement. Let S T be the
locus of the centres of gravity of the waterplanes. Through S, the point of
intersection of this locus and the line W L, let W. Z L 2 be drawn parallel to the

Fig. 237.

I—

base. The displacement to the waterplane W 2 L 2 will be very nearly equal to that
to the original waterplane W L. At amidships draw Q R normal to the keel
line. Q is the mean draught corresponding to W L , but this mean draught,
marked off on the displacement scale, would obviously give a reading less
than the actual displacement by the amount of the layer between W l L l and
W.,L 2 . The draught QR should therefore be employed.
Since the triangles W W X Q and SRO are similar—

qj? _ wj/i

w 1 o'

WW,
0R = TO"

RS.

But W Wi is half the trim, W l Q half the vessel's length, and RS the distance
abaft amidships of the centre of gravity of waterplane W»L>. Thus /?, the
amount to be added to the mean draught to get the draught to use with
the displacement scale, is readily obtained.

In an actual case, if RS were 4 feet, the length of vessel 360 feet,
and the trim 6 feet by the stern, we should get —

APPENDIX A.

299

180

x 4 x 12

of an inch,

corresponding to an increase of displacement over that given by the mean
draught of about 30 tons.

PROOF OF FORMULA BM = p.— In this formula, which expresses the

height of the metacentre above the centre of buoyancy—

BM = Height of metacentre (transverse or longitudinal) above centre
of buoyancy.
/= Moment of inertia of the waterplane about the middle line
as axis in the case of the transverse metacentre, and about
a transverse axis through the centre of gravity of the water-
plane in the case of the longitudinal metacentre.
V = Volume of displacement.

Fig. 238.

Consider first the transverse metacentre. Fig. 238, which illustrates the
case, is a transverse view of a vessel inclined through a small angle from
the upright. Before the inclination took place, the centre of buoyancy was
at B • it is now at B lt and has thus travelled the distance B B v The line
of the resultant upward pressure passes through B x and intersects the middle
line in M, which by definition is the transverse metacentre.

In the act of heeling the wedge of displacement WSW 1 passes across
the ship into the position LSU its centre of gravity moving from g x to g.,
in a line parallel to B B v If V be the volume of the ship's displacement,
and u the volume of either wedge—

V x BB X = u x g y g 2 ,
or l/xBI\/lx0 = ux g x g* (1)

Where 9 is the circular measure of the angle of inclination, which is as-
sumed to be very small.

300 SHIP CONSTRUCTION AND CALCULATIONS.

1/ is assumed to be known, so that to find B M it is only necessary
to calculate the value of the quantity u x g Y g,, i.e., the moment due to the
movement of the wedge of displacement across the ship. being small, S,
the point of intersection of the water lines W L and W X L^ is in the middle
line.

Calling the half breadth of the waterplane amidships, 6,

6 2

Sectional area of wedge W S W l or LS L x = —.ft

and Volume of a thin slice of either wedge = ~*ft 5 -*,

8X being the thickness of the slice.

Also Moment due to movement of thisl b 2 4 ,

volume across the ship ) 2 u 3

= * b\0.dX.

Now the moment of the whole wedge is equal to the sum of the moments of
all the slices into which it may be supposed divided. That is —

Moment due to movement of 1 x , 2 , , t .

u 1 a f = ^-b J .0.5x.

whole wedge ; 3

2 ...

But S-6°5x is the expression for the moment of inertia of the waterplane

about the middle line as axis. Calling this /, we get —
Moment due to movements
of wedge J

Substituting this for u x g } g 2 in (1) —

1.6 = V.BM.O

or, BM= T -

Take now the longitudinal metacentre. The inclination here is a fore-
and-aft one, but except as modified by this circumstance, the proof is the same.

Fig. 239 shows a fore-and-aft view of the vessel with a slight inclina-
tion aft. B is the centre of buoyancy when floating at the waterplane W\L U
Bi its position when at the line W L, ./??,* the intersection of the verticals
through B and B ls being the longitudinal metacentre ; 0, the projection in the
plane of the paper of the line of intersection of the waterplanes W L and
WiLu is called the centre of flotation and occurs at the same point in the length
as the centre of gravity of the waterplane W L h ly h 2 are the centres of the
immersed and emerged wedges. As in the previous case, we have —

1 ix h,h,= V x BB,
U being the volume of either wedge, and V that of the displacement.

The inclination being very small, B B x = B m x ft So that —
u x hih % = V x B m x 9 (2).

used here instead of ll/l to distinguish the longitudinal metacentre from the transverse.

APPENDIX A.

3 OI

To calculate the quantity u x h x h^ consider a small element of the
volume of the emerged wedge distant x from 0. The thickness of this
element is X x 0, and if y be the breadth of the vessel at the place, and
5 X the dimension of the element in the direction of the vessel's length, its
volume will be —

x.y.0.sx
and its moment about a transverse axis through

x 2 .y.0.dx.
The moment of the whole wedge is the sum of the moments of the

Fig. 239.

1 TO

elements, or

^xly.Q.sx,

and the moment of both wedges, double this quantity, or

2'2x\y.0.$x.
But 2^x i iy.dx is the moment of inertia of the watcrplane about a transverse
axis through its centre of gravity 0. Calling this I, and substituting the
value for the moment of the wedges thus obtained in (2), we get —

LB = V.Bm.Q

or,

Bm =

J_
V

CO-EFFICIENTS OF FORM.— These are useful in comparing one vessel
with another. The following are the usual co-efficients employed : —

302 SHIP CONSTRUCTION AND CALCULATIONS.

i, Co-efficient of area of load- water plane.

2. Co-efficient of area of immersed midship section.

3. Block co-efficient.

4. Prismatic co-efficient.

1. Co-efficient of Area of Load-Waterplane, — This is the ratio of
the area of the load-waterplane to that of a rectangle enclosing it. In a new
design it is important to know how this ratio compares with the corresponding
ratio of a vessel of known performance.

Example. — A vessel 360 feet long, 48 feet broad, is used as a basis in
designing another vessel 340 feet long and 46 feet broad. The load-waterplane
area in the standard case is 13,000 square feet, and it is intended to give the
new vessel the same co-efficient of load-waterplane. Calculate the area of load-
waterplane in the latter case.

1 3000

The standard co-efficient is — : 7: = '7 ^2: the area of the load-waterplane

360 x 48 /J v

in the new vessel will thus be : —

340 x 45 x '752 = 11505 square feet.

The load-waterplane co-efficient is also useful in approximate calculations like
the following : — A vessel of 330 feet length and 45 feet breadth floats at her
load draught. If 150 tons of cargo be discharged from a compartment amid-
ships, calculate the decrease in draught, assuming the co-efficient of the plane
of flotation to be "83, and the vessel to rise to a parallel waterplane.

Area of L.W.P. = 330 x 45 x '8$ = i2325"5 square feet.

Tons per inch immersion = — = 20^.

^ 420 y °

.'. Decrease in draught = — — = S'12 inches.

2. Co-efficient of Immersed Area of Midship Section. —This is the
ratio of the area of immersed midship section to that of a rectangle having
a depth equal to the vessel's moulded depth, and a breadth equal to the
breadth of the vessel. Thus, in the case of a vessel of 28 feet breadth and 8
feet mean draught, which has an area of immersed midship section of 210
square feet, this co-efficient is —

210

The immersed midship section co-efficient, like the previous one, is useful
in designing, and should be carefully considered where speed is an important
condition.

3. Block Co-efficient. — This is a volume ratio and expresses the
relation between the immersed volume of a vessel's body and that of
a rectangular figure surrounding it.

APPENDIX A. 303

If L = length of vessel,

B = breadth of vessel,

D = draught of vessel,

rr.i ™ , rr • volume of displacement

Inen, Block co-efficient = -, n n .

L x B x D

Example. — A vessel 500 feet long, 57 feet broad, has a moulded
draught of 28 feet. Calculate the displacement, assuming a block co-
efficient of 76.

Displacement = 500 x 57 x 28 x 76

= 606480 cubic feet.

Again, find the block co-efficient of a vessel, given the following par-
ticulars: — Length 185 feet, breadth 26 feet, mean draught in salt water
10 feet, displacement, 1000 tons.

„, . 1000 x 35

Block co-efficient = -5 ? =='727.

185 x 26 x 10 ' '

This co-efficient is of great value in comparing the forms of vessels,
but it must be used with care. It is easy to show that two vessels of
the same dimensions, block co-efficient, and displacement, may be very
different in shape. In one case the midship section may be full and the
ends fine; in the other, the midship section may be fine and the ends
full. Generally speaking, in cargo boats having large block co-efficients,
any fining of the body that is done should be reserved for the ends,
the midship section being kept full. "Where this has not been done,
vessels hard to drive and difficult to steer have resulted.

4. Prismatic Co-efficient. — This expresses the ratio of the volume
of displacement to the volume of a prism, whose section is the vessel's
immersed midship section, and length the length of the vessel. Thus, in
the case of a vessel 140 feet in length, which has an immersed midship
section area of 210 square feet, and a displacement of 640 tons, the
prismatic co-efficient is —

640 x 35 ,„ A

= 76.

140 x 210

It will be readily seen that this co-efficient affords a closer means of
comparing immersed forms than the block co-efficient, and in the case of
the two vessels above referred to, would show the one of fine midship
section and full ends to be of poor design, the prismatic co-efficient being
relatively higher than in the other vessel. It should be observed that a
relation exists between co-efficients 2, 3, and 4.

If y be the volume of displacement, A the area of immersed midship
section, G% G 3t C 4i the co-efficients 2, 3, and 4 above described, then —

304 SHIP CONSTRUCTION AND CALCULATIONS.

x B x

c 4

V
x A

= C<

, x B x

Now,

n _ __ _

Lx B x D x G 2

So that if any two of the foregoing co-efficients be known, the third can be
obtained.

It is usual to plot curves in the displacement scale diagram, showing
how the above co-efficients vary with change of draught From these curves
the co-efficients at any draught may be obtained by simple measurement.

APPENDIX B.

Table of Natural Tangents, Sines, and Cosines.

Angle

Tangent.

Sine.

Cosine.

Tangent.

Sine.

Cosine.

Tangent.

Sine.

Cosine.

Degs.
O

Degs.

— .

I'OOOO

•I763

■1736

•9848

•3640

•3420

'939 6

•OO43

■9999

I0 i

■1808

•1779

•9840

2 \

•3689

'346l

'9381

•0087

'9999

10J

•1853

■1822

■9832

2 \

•3738

'35 02

•9366

•OI30

'9999

1 of

■1898

•1865

'9824

2 of

•3788

•3542

'935 1

•0174

•OI74

•9998

' l 943

'I908

'9816

■3838

•3583

'9335

•02l8

"02l8

'9997

•1989

•195°

'9807

«i

•3888

•3624

•9320

'0261

'026l

■9996

•2034

* T 993

'9799

»i

'3939

•3665

*93°4

°3°5

'0305

*999S

III

•2080

'2036

■9790

2 If

•3989

*37°5

■9288

'°349

*°349

*9993

•2125

'2079

•978l

•4040

•3746

•9271

•0392

■9992

»i

'2171

•21 2 1

■9772

22^

•4091

■3786

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3°5

306

SHIP CONSTRUCTION AND CALCULATIONS.

Angle
in

Tangent.

Sine.

Cosine.

Angle
in

Tangent.

Sine.

Cosine.

Angle
in

Tangent.

Sine.

Cosine.

Degs.
3°

"5773

Degs.

*5000

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7372

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■3867

APPENDIX B.

307

Angle

Tangent.

Sine.

Cosine.

Tangent.

Sine.

Cosine.

Tangent.

Sine.

Cosine.

Deg-s.

Degs.

Deijs.

674

2*4142

•9238

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37320

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7'5957

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229*1816
Infinite

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•0000

Weights of Materials used in Shipbuilding.

Material.

Per Cubic Foot.

Materia'.

Per Cubic Foot.

Steel

490 lbs

Oak, Danzic

50 lbs.

Wrought Iron

480 „

Elm, English

35 ..

Cast Iron

45° »

Elm, American

44 u i

Gun Metal

534 ,*

Mahogany

53 .,

Brass, Cast

518 „

Greenheart

64 «

Lead

712 ,,

Ash

46 »

Tin

462 ,,

Teak

52 „

Zinc, Sheet

449 »

Pine, White -

35 »

Copper

549 »>

„ Red

36 „

Aluminium, Cast -

160 ,,

„ Yellow

3° »

Oak, English

58 „

„ Pitch

45 »

3 o8

SHIP CONSTRUCTION AND CALCULATIONS.

Rates of Stowage."

Cargoes.

No. of Cubic
Feet to 1 Ton.

Remarks.

Coal, Scotch

Coal, Welsh

Coal, Newcastle

Manchester Bales

5°

The figure may reach 160.

Pig Iron

/Stowed with as little wood pack-
\ ing as possible.

Alkali in Casks -

Wheat

Varies from 40 to 52.

Flour

Maize

Barley

Oats -

Varies from different causes ,
weight placed in each bag,

Cargo, rice in bags

amount of paddy, etc. Cargo
rice generally contains 20 per
cent, paddy.

Tea

83-120

Raw Sugar in Baskets

5°

Cotton, American

130

Cotton, Indian

Machine pressed.

Cotton, Egyptian

70-220

Jute

49-77

/The closer being very much
\ pressed by hydraulic power.

Wool, undumped

2 35

Wool, washed and dumped

100

Wool, greasy and dumped

Potatoes

5°

Bacon and Hains in cases

Peas and Beans

43-53

Beef, frozen and packed

9°"95

Beef, chilled and hung in quarters

120

Mutton, New Zealand

105-110

j Mutton, River Plate

115

In the above table of Stowage Rates no attempt is made to allow for
broken stowage, the figures being obtained from measurements of parcels
where the lost space was little or nothing.

From a paper by Professor Purvis in T.I.N. A. Vol. 26.

APPENDIX C.
Additional Questions.

i. Define the term area as applied to a plane surface,
area of the plate shown in the following sketch : —

Calculate the

Ans. — ^'75 square feet.

2. Calculate the area in square feet of the following: — (i) A square of
1 1 *5 feet side. (2) A rectangle of 15 feet length, 3I feet breadth. (3)
Triangle 6-5 feet base, 8-25 feet height. (4) A circle of 12-25 ^ eet diameter.

Ans.— (1) 132*25. (2) 56*25. (3) 26-81. (4) 117-85.

3. The ordinates in feet of a plane curve are 3, 5-5, 7-5, 8, and 9
respectively, the common interval being 8 feet. Between the first and second
ordinates, a half ordinate 4*6 feet is introduced, and another of 8*6 feet
between the fourth and fifth ordinates. Calculate the area in square feet.

Ans. — 220*4.

4. State the Five-Eight Rule ; upon what assumption is it based ?
Show how the Five-Eight Rule and Simpson's First Rule may be combined.

The half ordinates in feet of a portion of the load waterplane of a vessel

are 3j !•> 8, 8*5, 6'^, and 5 respectively, and the common distance between

them, 12 feet. Calculate the area in square feet, employing a combination

of the Five-Eight Rule and Simpson's First Rule.

Ans. — 822.

5. Referring to question No. 1, if the plate be steel $ of an inch

thick, what is its weight in lbs.?

Ans. — 937 lbs.

6. A solid wrought iron pillar, 18 feet in length, is 4^ inches in

diameter. Find its weight.

Ans. — 851 lbs.

* Many of these examples are based on questions set at the Board of Education Examinations
in Naval Architecture.

309

310 SHIP CONSTRUCTION AND CALCULATIONS.

7. A portion of a cyclindrical steel shaft tube, \\ inches thick, is 20
feet long, and its external diameter is 16 inches. Calculate its weight.

Ans. — 4646 lbs.

8. A deck 9000 square feet in area is to be laid with pitch-pine
planks 4 inches thick and 5 inches wide. There are two openings 16 feet x
12 feet and one opening 24 feet x 12 feet, which are not to be covered.

Calculate

(1) The number of running feet of deck planking;

(2) The weight of the wood deck, excluding fastenings.

Ans. — (1) 19,987. (2) 124,920 lbs.

9. A derrick post 18 inches external diameter is built of J-inch steel
plates. Estimate the weight of a length of 15 feet, neglecting straps and rivets.

Ans. — 1402 lbs.

10. Define displacement. The areas of the vertical transverse sections
of a ship up to the load waterplane in square feet are respectively 25,
105, 180, 250, 295, 290, 235, 145, and 30, and the common interval between
them is 20 feet. The displacement in tons before the foremost section is
5, and abaft the aftermost section is 6. Find the load displacement in cubic
feet and in tons (salt water).

Ans. — 30,900; 882 '8.

n. — Deduce a formula by which the tons per inch immersion at any
draught may be ascertained.

Given that the half ordinates in feet of the load waterplane of a vessel
are respectively -2, 4, 8-3, 11*3, 13*4, 13*4, 10-4, 7*2, and 2*2, and the length
of the plane 130 feet, calculate the tons per inch immersion in salt water.

Ans. — 5 '42.

12. A prism of rectangular section 120 feet long and 30 feet broad,
floats at a draught of 15 feet. Calculate the displacement in tons in salt
water; also construct the curve of displacement and the curve of tons per
inch of immersion for this vessel.

Ans. — 1543.

13. The tons per inch at the successive waterplanes of a vessel, which
are rj feet apart, are respectively 6'5, 6 - 2, 5 "6, 4*5, and o. Construct the
curve of tons per inch on a scale of 1 inch to 1 foot of draught, and 1
inch to 1 ton.

14. How is a "deadweight scale" constructed? Of what use is it to
the commanding officer?

15. What is meant by "mean draught"? Show that in the case of a
vessel floating considerably out of her normal trim, it is incorrect to use the
mean draught in reading the displacement from the displacement scale.

A vessel 300 feet in length floats in salt water and trims 8 feet by the
stern. If the waterplane intersects the locus of the centres of gravity of
waterplanes at a point 3 feet abaft amidships, measured parallel to top of

APPENDIX C

311

keel, and the tons per inch immersion be 23, estimate the difference between
the actual displacement of the vessel and that obtained from the displace-
ment scale, using the mean draught.

Ans. — 22 tons.

16. Obtain an expression giving the extent to which a vessel rises in
passing from fresh to salt water.

A vessel whose displacement is 4000 tons, leaves a harbour in which
the water is partly salt, and proceeds to sea. If the water in harbour weighs
1 o 1 5 ozs. per cubic foot, calculate the number of inches through which
the vessel will rise on reaching salt water, given that the tons per inch is 30.

Ans. — 1 '18.

17. A vessel of box form is 210 feet long, 30 feet broad, and has an
even draught of water of 10 feet when floating in sea water. If a sheathing
of teak 3 inches thick were worked over the bottom, and also over the ends
and sides to a height of 12 feet above the bottom, what would be the
additional weight, taking teak at 50 lbs. per cubic foot, and what would then
be the draught of water.

Ans. — 68 tons, 10 feet z\ inches.

18. A vessel carries in her hold a cube, each side of which is 10
feet. If the cube be put overboard and attached to the ship by means
of a chain, what will be the effect upon the vessel's draught, the cube being
supposed of greater density than salt water. The area of the vessel's water-
plane is 4000 square feet.

Ans, — Vessel rises 3 inches.

19. A rectangular pontoon 100 feet long, 50 feet broad, 20 feet deep,
is empty, and floating in sea water at a draught of 10 feet. What altera-
tion will take place in the floating condition of the pontoon if the centre
compartment is breached and in free communication with the sea, if —

(a) The pontoons were divided into five equal watertight com-

partments by transverse bulkheads, extending the full depth of
the pontoon ?

(b) The watertight bulkheads stopped at a deck which is not water-

tight, 12 feet from the bottom of the pontoon ?

((a) Vessel will sink bodily 2 feet 6 inches.
Ans.-

1(b) Vessel will founder.

II.

1. Show how the principle of moments is applied in obtaining the
centre of gravity of a plane area such as a vessel's waterplane.

2. The half-ordinates of a load-waterplane of a vessel in feet, com-
mencing from aft, are, respectively— -i, 5, ri'6, 15-4, i6'8, 17, 16-9, 16-4,
I 4'5) 9"4» an( * 'h anc * tne common interval is n feet. Find —

312 SHIP CONSTRUCTION AND CALCULATIONS.

(i) The area of the plane in square feet;

(2) The distance of its centre of gravity from the 17-feet ordinate,

stating whether the centre of gravity is before or abaft that

ordinate.

Ans. — (1) 2732*4. (2) 2'83 feet forward.

3. A vessel is 180 feet long, and the transverse sections from the
load-waterline to the keel are semicircles. Find the longitudinal position of
the centre of buoyancy, the half-ordinates of the load-waterplane being 1,
5, 13, 15, 14, 12, and 10 feet, respectively.

Ans. — 73*76 feet from 10-feet ordinate.

4. Given a diagram showing the locus of the centre of buoyancy,
constructed as described in Chapter II., explain how the height of the
centre of buoyancy corresponding with any waterplane may be ascertained.

5. Construct the locus of the centre of buoyancy for an upright
prism of rectangular section, and also for a prism whose section is an
equilateral triangle, and which floats with one of its faces horizontal.

6. Illustrate by a simple example the arrangement of the numerical
work usually followed in an ordinary displacement paper for obtaining the
displacement and position of the centre of buoyancy of a ship.

7. The load displacement of a ship is 5000 tons, and the centre of
buoyancy is ro feet below the load-waterline. In the light condition the
displacement of the ship is 2000 tons, and the centre of gravity of the
layer between the load and the light lines, is 6 feet below the loadline.
Find the vertical position of the centre of buoyancy below the loadline in
the light condition.

Ans. — 16 feet.

IV.

1. Distinguish clearly between hogging and sagging strains. What causes
these strains, and at what parts of a loaded cargo steamer are they likely
to be a maximum ?

2. What is a u curve of weight," and a "curve of buoyancy"? De-
scribe how these curves are constructed for a vessel afloat in still water.
What conditions must these curves comply with in relation to each other?

What are the usual assumptions made in constructing curves of weight
and buoyancy for a vessel afloat among waves?

3. A vessel of box form 240 feet long, 40 feet broad, 20 feet deep,
floats in salt water at a level draught of 8 feet. If the vessel's weight is
1000 tons evenly distributed, and she is loaded at each end for length of
70 feet with 600 tons, also evenly distributed, draw the curves of weight
and buoyancy.

4. Write down the formula employed in calculating the longitudinal
stress on the material at any point of a section of a beam under a longi-
tudinal bending moment.

APPENDIX C. 313

5. What assumptions are made in applying the formula referred to in
the previous question to the case of a ship ?

6. A steel beam of X section is 12 inches deep, -| inch thick, and has
6-inch flanges top and bottom. Calculate the moment of inertia of the section.

Ans, — 254 inch units.

7. Referring to the previous question, if the beam is 20 feet long and
supported at each end, and loaded in the middle with a weight of 6 tons,
calculate the maximum tensile and compressive stresses in tons per square
inch. The weight of the beam itself may be neglected in working out the
problem.

Ans.—S'S, S-$.

8. State the maximum longitudinal stress, as ordinarily calculated, in a
large and in a small steel cargo steamer, when poised on waves of their
own length. Give reason for any difference in the values.

9. In the case of some large steel passenger vessels having long super-
structures of light build, the latter are cut about mid length, and a sliding
joint made. What is the reason for this?

10. Enumerate the stresses to which ships are subjected which tend to
produce changes in their transverse forms. State what parts assist the
structure to resist change of form.

1. Sketch and describe the three-deck, spar-deck, and awning-deck
type of vessels.

2. In designing a cargo vessel of full form, state generally how you
would proceed to shape the body with a view to securing the best results.

3. Sketch in profile a well-deck and a quarter-deck type of vessel.
What are the essential features of each type?

4. Describe briefly the trend of development in the construction of
cargo steamers. Sketch in section a vessel on the web frame system,
also one with deep frames.

5. Sketch in outline midship section of a turret-deck steamer. What
are the advantages claimed for this type over cargo vessels of ordinary form?

6. Describe with the sketches the Ropner Trunk Steamer, and the
Isherwood Patent Ship. What are the chief features of these types?

VI.

1. Sketch and describe an ordinary bar keel.

2. How is the scarph of a bar keel formed? What is the length
of a scarph in terms of the thickness of the keel? How are the rivets
arranged, and what is their spacing?

3. Before proceeding with the framing it is necessary to set the keel

314 SHIP CONSTRUCTION AND CALCULATIONS.

straight on the blocks, How are the keel lengths temporarily joined
together so that this may be correctly done?

4. What is a side-bar keel? Is it a better or worse form than that
referred to in the previous questions ? Give reasons.

5. Mention any practical difficulty attendant on the constructing of a
keel on the side-bar system, and state what means are taken to over-
come it.

6. What are the advantages and disadvantages of projecting keels ?
Sketch and describe a form of keel which entails no outside projection,
and show that the arrangement is satisfactory from a point of view of
strength.

7. Show a good shift of butts of the flat keel with reference to those
of the vertical keel and angles connecting them ; also with reference to the
garboard strakes of plating,

8. Describe, and show by sketches in section and side elevation, how
an intercostal plate keelson (or vertical keel) is worked and secured in an
ordinarily transversely framed vessel with a flat plate keel.

9. What are bilge-keels ? Why are they fitted ? Sketch an efficient
form of bilge-keel, indicating the connections to the hull.

1 o. Why are hold keelsons fitted ? Sketch a side and a bilge-keelson.
What advantage is gained by fitting intercostal plates to the shell between
the keelson bars ?

11. How is the strength maintained at the joints of hold keelson bars?
Make a sketch showing details of riveting, etc.

12. What is the usual spacing of transverse frames? Show by a sketch
how a frame, reverse frame, and floorplate are connected.

13. What are frame heel pieces? Where are they fitted? They are
not usually fitted at the ends of a vessel. Why ?

14. Sketch an ordinary floor. How far does it extend up the ship's
side? Where are floorplates usually joined? Make a sketch at a joint,
showing the rivets.

15. Show in section the common forms of ship beams, and state where
each section should be employed.

16. Why are beams cambered ? Is their strength increased thereby ?
What is the usual camber of upper-deck beams?

17. Deck beams are sometimes fitted at every frame, and sometimes
at alternate frames. State the circumstances in which each arrangement may
be employed to most advantage.

18. Why are deck beams not reduced towards their ends, as on the
principle of the girder they might be.

19. Describe the usual methods of forming "bracket," "slabbed," and
"turned" beam knees, and state which, in your opinion, is most efficient.

20. Sketch a bracket knee showing in detail the connections to the
frame and beam. What are Lloyd's requirements as to the number of rivets
in beam knees?

APPENDIX C. 315

21. In the design of a certain vessel, requiring by rule a tier of
lower-deck beams at the usual spacing, it is proposed to modify or dispense
with the latter in order to improve the facilities for stowage. Show how
this might be done without reducing the strength.

22. What are web frames? Why are they fitted? Sketch a web frame
showing all connections in a vessel having ordinary floors.

23. Discuss the relative merits of making the web frames continuous,
and hold stringers intercostal, and vice versa. Show in detail the connec-
tion of a web frame to the margin-plate of an inner bottom.

24. What is meant by "deep framing"? What are the advantages of
this system of construction over that consisting of combined ordinary fram-
ing and web frames.

25. Sketch and describe a M'lntyre ballast tank. What are its essential
features ?

26. Describe the cellular system of constructing double bottoms; com-
pare it in details with the system referred to in the previous question.

27. Assuming continuous longitudinals and intercostal floors, show by
sketches the construction of a cellular double bottom for a length of one
compartment, indicating the man-holes through the longitudinals and tank
top, and showing details of the connections of the longitudinals to the floors,
tank top and shell.

28. Show by a sketch how the plating of the tank top or inner bottom
is usually arranged, giving details of the butt and edge connections. At certain
parts the plating is increased in thickness. Name these parts, and state why
the increase is made.

29. Signs of straining have frequently been observed in the riveting
connecting the tank knees to the margin-plate of the double bottom, par-
ticularly at the upper part of the knees. Show by a sketch the means
usually taken in modern vessels to prevent such straining.

30. What are the advantages and disadvantages of flanging the edges
of plates in lieu of fitting angles ?

31. Why are wash-plates fitted in deep ballast tanks and in peak tanks?

32. Show by a sketch how a deep tank is made watertight at the deck.

33. What considerations determine the diameters of pillars in a ship?
In fitting pillars to beams, where should they be placed in order to develop
their greatest efficiency?

34. What is the limit of breadth of ship allowed by Lloyd's Rules
for one and for two rows of pillars, respectively?

35. Show by sketches the usual methods of attaching pillars at their
heads and heels.

36. In the case of a deck having beams at every frame, show how it
may be efficiently supported by a tier of pillars at alternate frames.

37. Sketch an arrangement of wide-spaced pillars, showing how the deck
between the pillars is supported. Give details of the attachments to tank
top and deck.

316 SHIP CONSTRUCTION AND CALCULATIONS.

38. Certain parts of the shell-plating of a ship are thicker than others.
Name these parts, and give reasons for the increased thickness.

39. What are Lloyd's requirements regarding the length of shell-plates
and the position of end joints? Sketch a good arrangement of shell butts
or joints.

40. Sketch and describe the various plans adopted of fitting shell-
plating, indicating specially a system by which the fitting of frame packing
pieces is obviated.

41. What are Lloyd's requirements as to the number of rows of rivets
in shell landings? Show by rough sketches a single, a double, and a
treble-riveted edge lap, indicating the thickness of the plates, width of laps,
and diameter and spacing of rivets.

42. What are the advantages and disadvantages of overlapped end
joints as compared with butted joints ?

43. Lapped joints and butted joints having single straps, show a
tendency to open when under stress. What is the reason of this ?

44. How would you proceed to stop a leaky end joint of butted type
in a strake of bottom plating?

45. It is the practice in many shipyards to scarph overlapped joints
where they are crossed by the landings so as to avoid the use of packing
pieces. Show by sketches how this is done (a) in the case of a joint in
an outside strake ; (b) in the case of a joint in an inside strake.

46. In a riveted joint, discuss the general considerations which govern
the diameter and pitch of rivets, and their distance from the edge of the
joint.

47. Explain why, as a rule, the ratio of the size of rivets to the
thickness of the plates they connect becomes reduced as plates increase in
thickness. Show that a limit to this ratio is fixed by practical considerations.

48. Sketch and describe the various heads and points common in
ship work, and state where each is used.

49. State the diameters of rivets required by Lloyd's Rules for plates
of the following thicknesses — £", V, f", and 1", respectively. What should
be the pitch of rivets in watertight work?

50. What is the spacing of rivets in frames and beams? Why is the
rivet spacing closer in bulkhead frames than in frames elsewhere, and how
is the loss of strength thus caused made good?

51. Why are the rivets connecting the framing to the shell-plating of closer
pitch in way of deep water ballast tanks and peak ballast tanks than elsewhere.

52. Rivets are usually manufactured of cone shape under the heads.
Why ?

53. Two plates have to be joined by rivets. Discuss the advantages
and disadvantages of —

(a) Punching the rivet holes ;
(&) Drilling the rivet holes.

APPENDIX C. 317

Describe how the punching and fitting of the plates should be con-
ducted to secure efficient work.

54. Iron rivets are found to have a higher strength efficiency in iron
plates than in steel plates. Give a reason for this. Why are iron rivets
employed in steel shipbuilding in preference to steel rivets ?

55. What is a drift punch? Explain its uses., Show that in certain
circumstances the use of a drift punch might lead to bad workmanship.

56. What are the principal functions of a deck stringer ? Show by
sketches how you would connect and^ fasten a stringer to the beams, fram-
ing, and plating of a ship.

57. How would you proceed in arranging the fastenings in a stringer
plate at the butts? A stringer plate is 50 inches wide and -|-inch thick;
sketch the riveting in a beam and at a butt, and show that the arrange-
ment is a good one.

58. A steel ship is found on her first voyage at sea to be structurally
weak longitudinally. How would you attempt to effectually strengthen the ship
with the least additional weight of material?

59. What are deck tieplates? Sketch an arrangement of tieplates on
the main deck of a sailing-ship, showing how they are fitted. Explain why
they are arranged diagonally as well as fore-and-aft.

60. Decks require to be strengthened in way of large openings.
Show by a sketch the usual compensation at the sides and corners of a
large upper-deck cargo hatch.

6 1 . Discuss the relative values of teak, pitch pine, and yellow pine,
as materials for deck planking.

62. Describe in detail how you would proceed to lay a wood deck

(a) Where no steel deck is fitted ;

(b) Where there is a steel deck.

Show by sketches the connections at a butt joint of the deck planking in
each case.

63. What is the Rule height for hatch coamings at upper and at bridge
decks. Show by detail sketches how the end and side coamings of an upper-
deck hatchway are bound to the deck structure.

64. How are hatch openings protected against inroads from the sea ?
Sketch an arrangement of beams for supporting the covers of a main cargo
hatch in a modern vessel.

65. Describe the mechanical appliances usually installed in cargo steamers
for loading and discharging cargo.

66. Sketch a derrick, showing how it is supported at the heel, and
detail the arrangements for topping and slewing it.

67. Assuming two winches to be fitted to one hatch, sketch roughly
two arrangements by which direct leads to the winch barrels may be obtained.

68. In what circumstances may it be desirable to hinge the derricks
on special posts instead of on the masts? Sketch a derrick-post and
derrick, and show how the former is connected to the deck.

318 SHIP CONSTRUCTION AND CALCULATIONS.

69. How are steam winches supported (a) on an unsheathed steel deck?
(b) Where a wood deck is laid? What arrangement is made to minimise
vibration of the deck due to the working of the winches ?

70. Show in section the construction of a lower mast in a large
sailing-ship. At what parts is the mast-plating doubled? Why are the
doublings fitted?

71. Show by rough sketches how a mast is wedged at a deck, and
how it is supported at the heel

72. Sketch an appliance fitted in modern vessels for tightening up the
standing rigging. Show how it is connected to the ship.

73. What is a "spiked bowsprit"? Show how a bowsprit is sup-
ported and stayed.

74. What reduction in diameter is allowed in a steamer's masts as
compared with those of a sailing-ship? How is a steamer's mast supported
at the heel where it is stopped at a lower deck ?

75. State the advantages of having a good system of watertight bulk-
heads in a steamer. What are Lloyd's requirements in respect to water-
tight bulkheads for a steamer of 300 feet and one of 400 feet length,
respectively ?

76. Explain why, in ordinary cases, only one transverse watertight bulk-
head is fitted in a sailing-ship.

77. How are watertight bulkheads usually built and stiffened? Show by
a sketch the arrangement of the plating, spacing of stiffeners, and details of
the attachments of the latter in a main transverse watertight bulkhead of a
large cargo steamer.

78. In many modern cargo steamers the stiffeners below the deck are
fitted vertically only. What are the advantages of the arrangement ? Are
there any disadvantages ?

79. Taking the case of a fore-peak bulkhead, which is deep and
narrow, how would you arrange the stiffeners so as to get the greatest
efficiency with the least weight of material?

80. A hold stringer consisting of a bulb plate and double angles passes
through a watertight bulkhead. Show how you would make the bulkhead
watertight around the girder.

81. Referring to the previous question, if the stringer were stopped on
each side of the bulkhead, show by a sketch how you would endeavour
to maintain the strength at the junction.

82. Sketch and describe a common method of fitting a stem bar in a
modern cargo vessel, where there is a flat plate keel.

83. Sketch roughly an iron or a steel sternpost of a cargo steamer,
showing how it is connected with and fastened to the keel. Why is the
sole piece of the sternpost of a single-screw ship frequently made broad and
shallow in way of the aperture?

84. Sketch a a bracket arrangement as fitted for supporting the after-
end of each propeller shaft in a small twin screw steamer.

APPENDIX C. 319

What is the principal objection to a brackets? Describe a plan by
which this is overcome in many modern high speed vessels.

85. Sketch and describe a modern single plate rudder, showing the
spacing of the arms and details of the pintles.

86. Commonly, a rudder is supported by the bottom gudgeon of the
sternpost. Sketch the arrangement and indicate the means taken to ensure
that the rudder shall work without undue friction.

87. Show by rough sketches the usual method of preventing the ac-
cidental unshipping of a rudder and of limiting the angle through which
the rudder turns.

88. Sketch a rudder coupling, the diameter of rudder stock being 9
inches ; indicate the number, position, and diameter of the bolts.

VII.

1. Define stable, unstable, and neutral equilibrium as applied to the
case of a vessel floating freely in still water. Illustrate your definitions by
suitable sketches.

2. Explain briefly what are the elements in the design of a vessel
which control the position of the transverse metacentre. Show that the position
of the metacentre is only of relative importance.

3. Describe in detail an "inclining experiment." State what precautions
should be taken in order to ensure a reliable result.

An inclining experiment is to be conducted on a certain vessel, her
displacement at the time being 2600 tons, and mean draught 8 feet 6 inches.
The inclining weight is 6 tons, arranged in two lots of 3 tons, one on each
side of the upper deck. The pendulum is 29*5 feet in length. The follow-
ing is done : — First, one lot of the inclining weights is moved from port to
starboard through 40 feet. The deflection of the pendulum is observed and
the weight returned to its original position. Then the second lot is moved
from starboard to port through the same distance, an observation taken, and
the weight, as before, returned. The mean deflection of the pendulum is
found to be 1*9 inches. Estimate from the information given the metacentric
height of the vessel when in the above condition.

A/is.S'6 feet.

4. Obtain and prove the expression for the height of the transverse
metacentre above the centre of buoyancy.

5. A vessel is 30 feet wide, 15 feet deep, and the centre of gravity
of the vessel and its lading is at the middle of the depth of the vessel
for all variations in the draught of water. Construct to scale the metacentric
diagram.

6. Sketch the metacentric diagrams of any two vessels of different types
with which you are acquainted. Give reasons for any differences * in the
form of the curves.

320 SHIP CONSTRUCTION AND CALCULATIONS.

7. A vessel 140 feet long, and whose body plan half sections are squares,

floats with its sides upright, and the centres of all the sections lie in the

plane of flotation. The lengths of the sides of the sections, including the

end ordinates, are '8, 3*6, 7*0, 8*o, 6*4, 3-0, and 7 feet, respectively, the sections

being equally spaced. Calculate the distance between the centre of buoyancy

and the metacentre.

Am. — 4*51 feet.

8. In the case of what class of vessel must the centre of gravity be
below the centre of buoyancy, for equilibrium?

9. A vessel of constant rectangular section, 200 feet long, 40 feet broad,
draws 20 feet of water when intact. Two rectangular watertight compartments,
10 feet in width, measuring in from the ship's side, and 10 feet in depth,
the bottom of each being 6 feet below the original waterplane, extend each
side of amidships for a length of 60 feet.

If the centre of gravity of the vessel is 15 feet above the keel, find
the metacentric height — (a) When the vessel is intact (b) When the side
compartments (assumed empty) are in open communication with the sea.

Ans.-{ {a) r66 feet
\{b) -07 feet.

VIII.

1. Obtain the expression which gives the height of the longitudinal
metacentre above the centre of buoyancy.

2. Calculate the longitudinal metacentric height for a log of wood 20
feet long and of square section, the side being 2 feet 6 inches, when floating
freely and at rest at a draught of r foot 6 inches.

Ans. — 2172 feet.

3. A raft 15 feet long is constructed of two logs of timber 18 inches
in diameter and 4 feet between centres, and is planked over with wood
3 inches thick, forming a platform 12 feet by 5 feet. All the wood is of
the same density, and the raft floats in sea water with the logs half immersed.
Find the longitudinal metacentric height and the moment to alter trim 1 inch.
{See note to question No. 6 on opposite page).

Ans. — 31*33 feet, 295 foot lbs.

4. A small weight is placed on board a vessel in any longitudinal
position. Explain how you would proceed to find the changes in the draughts
forward and aft.

5. A cargo vessel is 48 feet broad on the load waterline. Given that
the tons per inch of immersion is 35, calculate approximately the moment
to alter trim 1 inch.

Ans.~ 788 foot tons.

6. A vessel of circular section, 80 feet long and 20 feet diameter,

APPENDIX C. 321

floats with the axis in the waterplane. Calculate the trimming effect of
shifting a weight of 15 tons from mid length to a point 10 feet from the
after end. The centre of gravity is 2 feet below the waterplane.

Note. — The centre of buoyancy may be fixed in relation to the trans-
verse metacentre.

Ans. — i2>\ inches by the stern.

7. Describe any simple method of providing commanding officers with
such information concerning their own vessels as will enable them to deal
quickly and correctly with trim problems.

8. The trim line of a certain vessel corresponding to the load draught
makes an angle of 42 degrees with the horizontal.

Plot the trim line, and from it obtain the change of trim due to shifting
50 tons through 100 feet aft. The displacement is 8000 tons.

Ans. — 6| inches by stern.

IX.

1. Given that the righting levers of a vessel at angle of 15 , 30°, 45 ,
6o °j 75°> an d 9°° respectively, are ^74, 1*53, 2*1, 2*18, 1*65, '9 feet, con-
struct the curve of stability, and indicate the maximum righting lever and
the angle at which it occurs. The metacentric height is 2*62 feet.

Ans. — 2*22 feet, 55°.

2. What are the features in a vessel affecting the range of the curve
of stability? Show that a great metacentric height may be associated with
a short range.

3. Draw in one figure the curves of stability of two dissimilar types
of vessels with which you are acquainted, and give the reasons for any
differences which exist in the nature of the curves you show.

4. Some merchant vessels will not remain in an upright position when
unloaded. Explain the reason of this. Draw the curve of stability of a
vessel when in the condition named.

5. A sailing-ship is heeled by the pressure of the wind on the sails.
Assuming her to be at a steady angle of heel, show in a sketch the
forces acting, and state the relation of the moments of these forces to
each other.

6. A vessel of box form 200 feet long, 40 feet broad, 20 feet deep,
floats in sea water at a level draught of 15 feet. Assuming a metacentric
height of 2 feet, construct the curve of statical stability.

7. Draw cross curves of stability for a vessel of square section at
angles of 45° and 90 respectively, assuming the centre of gravity to be
1 foot below the centre of the section.

8. Having given the value of the righting arm of a vessel at a certain
inclination when at her load displacement, the position of the centre of

322 SHIP CONSTRUCTION AND CALCULATIONS.

gravity being known, show how you would find it at the same inclination
when at a reduced displacement, due to the consumption of the bunker coal.

i. What is meant by the phrase "Period of a single roll"? It is
desired to obtain the period of roll of a cargo vessel when in a given
condition. How could this be ascertained experimentally ?

2. What is the transverse radius of gyration ? How is it obtained ?

3. What effect has the variation of the metacentric height upon the value
of the rolling period?

In a given vessel what is the difference between the rolling periods
corresponding to a metacentric height of 2 feet and 4 feet, respectively,
assuming the transverse radius of gyration to be 18 feet and the same in
both cases ?

4. Explain why waves that are relatively high in relation to their lengths
are more powerful in causing vessels to roll heavily than waves that are
relatively low.

5. Describe a simple experimental method of proving that a vessel
when broadside on to a series of regular waves always tends to place her
masts parallel to the normal to the wave slope.

6. State the length and period as actually observed of large Atlantic
storm waves ordinarily met with.

What, by inference, should the natural roll period approximate to in
the case of a vessel intended to trade in the Atlantic, in order to obtain
the best results.

7. Mention an appliance that has been recently employed to minimise
the rolling motions of vessels at sea. Show by quoting the results in any
actual case, what success has attended the new system.

1. A vessel is to be loaded with a general cargo of which the weights
and other particulars are known. How would you proceed to find the
vertical position of the centre of gravity ? Assuming the displacement scale
and the diagram of metacentres to be available, how would you determine
the metacentric height with the proposed system of loading?

2. The stability curve at the load draught in a certain vessel is of
considerable area and range, but shows upsetting levers at angles near the
origin. How do you account for this? In the case of such a vessel, what
considerations would influence you in fixing upon a value of metacentric
height with which to start a voyage.

3. What is the chief objection to deck cargoes? Show that a deck
cargo of timber, if well stowed and secured, may improve a vessel's sea,
qualities.

APPENDIX C 323

4. What is the angle of repose for wheat?

In certain circumstances, grain carried in the hold of a steamer is
found to slide at a much smaller inclination than its normal angle of
repose. Describe these circumstances, and explain the causes to which they
give rise which lead to the reduction in the sliding angle.

5. What proportion of the full deadweight should a modern steamer
carry in making an Atlantic voyage in ballast? To what extent should the
propeller be immersed? What has frequently happened when a voyage has
been made in too light a trim, and rough weather has been encountered ?

NDEX.

Area, Centre of Gravity of
,, Metrical Units of.
,, of Portion of Curve between Two

Consecutive Ordinates
,, of Rectangle
,, of Rhomboid
,, of Square
,, of Trapezoid
,, of Triangle .
,, of VVaterplane
Areas, Combination of Simpson's Rules fo:
,, Simpson's First Rule for
,, Simpson's Second Rule for
,, Tchebycheft's Rule for . . I

,, Trapezoidal Rule for .
Atwood's Formula for Statical Stability
Awning Deck Vessels, Restriction of

Draught in
Balanced Rudder ....
Ballast, Advantages of Water over Dry
,, Conditions Fixing Amount Re

quired ....
,, Longitudinal Disposition of
,, Stowage of
Ballast Tanks, Early Methods of Con
structing

,, Function of . . 109

,, Methods of Making Water

tight joint at margin of
,, M'Intyre System of Con-

structing
,, of Special Type

,, Testing of

, , Why Seldom Fitted in

Sailing Ships
Ballasting, Danger of Filling Tanks at Sea
,, Effect on Stability of Filling

Tanks ....
„ Importance in Minimising

Pounding Strains of Efficient
,, Purpose of . . . 284

Bar Keel, Chief Objection to .
,, Description of

,, with Intercostal Centre Keelson

,, with Single Plate Centre Keelson

Beams, Function of Deck

,, Number of Tiers required in a

Vessel .....

Beam Knees, Bracket ....

,, Comparison of Methods of

Forming . . 10S,
,, Considerations influencing

Depth and Thickness of

2S
1

>>37

2
2
1

3
2
6
10
6
9

1-13
4

221

76
1761

2S6

2S5
280
2S5

1 10
■no

III

120

120
I 10

2S6

2S7

73
285

93
95
94
io5

105
107

109

Beam Knees, Width across Throats

Lloyd's Rules for Number
of Rivets in .

„ Size of in Single-Deck

Vessels .... log

Slabbed .... 107

,, Turned . . 108, 109

Beam, Effect of on Curve of Stability 237, 241

PAGE
IO9

109

IOO
I06
106

I06
I07

Beams, Lloyd's Rules for Spacing of
,, Method of fitting Wide-Spaced .
,, Reason for Giving Camber to
,, Under Unsheathed Steel or Iron

Decks

Beam Sections, Forms of

Bending Moments and Shearing Forces

of Floating Vessels . 51, 52, 55

Bending Moments and Shearing Forces

of Simple Beams .... 45-50
Bending Moments, Effect of Orbital

Motion of Water Particles on . . 54

Bilge Keels 96, 97

Bilge Keels, Experiments with H.M.S.

Repulse and Revenue . 265

,, Extinctive Value of 264, 265, 266

,, Prof. Bryan's Investigations 265

Bilge Keelson . ...

Block Co-efficient .....

Bowsprit, How Secured and Stayed

Bowsprit, Spiked .

Bow Rudder, design of .

Bow Rudder, Function of

Bulkheads, Arrangement and Spacing of

Stiffeners of

„ Centre Line .

,, Connection of to Ship's Side 169-170

,, Construction of . . 168-171

,, Function of . . .165

,, Lower Limit to Number of in

Steamers
,, Lloyd's Rules for Number of

Transverse Watertight
,, Screen. ....
,, Spacing of Rivets in Water-
tight. . . I37-I3S>
Stiffening of Peak Tank
,, Thickness of Plating of
,, Value of Collision
Bulkhead Liners, Compensation for Omit-
ting

Bulkhead Liners, Reason for fitting 138,
Bulkhead Stiffeners, Advantage of flang-
ing plates in lieu of-

104
302
163
163
176/
1760

16S
171

166

167
171

168

117
168
166

170
170

169
Bulkhead Stiffeners, Method of Fitting 16S, 169

3 2 4

INDEX.

325

I'AQB

Bulkheads of Deep Tanks . . 169
,, ,, Oil Vessels . . .169
,, Height of Transverse Water-
tight 167

Bntlslraps of Keelson and Hold Stringer

Angles, Method of Fitting . 104-105
Cargo Steamers, Awning or Shelter-
Deck Type of . 76, 78
,, ,, Compensation for Omis-

sion of Hold Pillars in 84
,, ,, Erections on . . 77

,, ,, Freak Designs of . 77

,, ,, Isherwood Type of . 89-92

,, ,, Partial Awning-Deck

Type of 79

,, ,. Quarter-Deck Type of. 78

,, ,, Single- Deck Type of . 83

,, ,, Strength Types of . 75

,, ,, Three- Island Type of. 77

., ,. Trunk Type of. . 87

Turret-Deck Type of . 85

Well-Deck Type of 77

Camber to Beams, Reason for giving . 106

Cargo Cranes, Advantage of . . . 155

Cargo Gear in Sailing-Ships . . . 155

Cargo Gear in Steam-Ships . . . 155

Cargo Hatchways, Arrangementsforsecur-

ing Water-tightness of 152
,, ,, Arrangements for tak-

ing Chafe of Cargo . 152
, y ,, Function of . . 148

, , , , Height of Coamings of 148-149

,, ,, Method of Framing . 148

,, ,, Size in Modern Vessels

of .... 148
,, ,, Web Plates and Fore

and Afters in . 151-152

Cargo Ports and Doors . . . 152-154

Cellular System of constructing Double

Bottoms 112-115

Centre of Buoyancy, Approximate Posi-
tion of . .184
,, ,, Explanation of Term 33
,, ,, Locus of . . 36, 40
,, ,, of Prismatic Vessels 33
,, ,, of Vessels of Or-
dinary Form . 34-40

Centre of Effort 249

Centre of Gravity, Definition of . 28

,, Influence on Curve of

Stability of Position of 240
„ of an Area ■ . . 28-32

„ of a Ship, Method of

Finding Position of 187-191
,, of the Area of a Half-

Waterplane . . 29-32
Centre Girder, Connection to Flat-Plate

Keel 9 6 > 97

Centre Keelson . . . . 44> 93"97
Centre-Line Bulkhead, Construction of . 171
Centre of Lateral Resistance . . . 249
Change of Draught in Passing from Fresh

to Salt Water 297

Change of Trim .... 198, 213
Coal Cargoes, Precautions Necessary in

Loading ... .281

I'AGIi

Co-efficient of Load Waterplane . . 302
„ Midship Section . . 302

, , Resistance to Roll j ng,

Froude's . . . 264
Collision Bulkhead, Stiffening of . 168, 169
,, Value of . . 166

Compressive Stresses, Effect of, on Thin

Deck-Plating .... 66

Correction of Wedges . . 225, 226

Countersinking of Rivet Holes . . 139

Coupling for Rudder . . 176/- 176'/*

Cross Curves of Stability . . 231-236

Cross Curves of Trim for Box-Shaped .

Vessel 211

Cross Curves of Trim, how Obtained . 209
Curve of Bending Moments and Shearing

Forces for Simple Beams . 46-50
,, Bending Moments and Shearing
Forces for Vessel Afloat i n
Still Water .... 51

,, Bending Moments and Shearing
Forces for Vessel Loaded with
Homogeneous Cargo . . 5 2

,, Bending Moments and Shearing
Forces for Vessel on a Wave
of Her Own Length . . 55, 56
„ Centres of Buoyancy . 36, 40, 185

,-, Displacement, how Constructed 20

,, Flotation .... 255

,, Loads for Simple Beams . . 49, 50

,, Loads of Vessel Afloat in Still

Water 51, 52

,, Loads of Vessel Among Waves 55, 56
,, Moment to Alter Trim One

Inch .... 199, 213

,, Tons per Inch Immersion . 22

,, Transverse Metacentres . 185, 186

Curves of Stability, Effect of Beam on 237, 241

,, ,, for Commanding

Officers . 289, 290

. , „ for Vessels of Cir-

cular Section . 218

,, ,, for Vessels of Box

Form . 237, 239, 240
,, ,, Influence of Free-

board on 238, 239, 241
,, ,, Influence of Posi-

tion of Centre of
Gravity on . . 240
,, ,, of Actual Ships . 241

,, ,, Tangent at the

Origin . . 230, 231
Curves of Weight and Buoyancy . . 54

Deadweight Scale ..... 21
,, ,, Use of to Ship's Officers 21

Deck Beams, Function of 105

Decks, Comparative Values of Wood and

Steel 142

,, Function of . , . .142

Deck Loads . . . . 281

,, Openings, Strengthening at Cor-
ners of ... 146
„ Plating, Compensation for Cutting

Openings in . . , 145

„ Plating, Objection to Joggling

Edge Seams of . . , 145

326

INDEX-

■ 145

44, 142
142
119

119
152

119
Il8

- 157
154-157

156

155

157
157
20

Deck Plates, Precaution Necessary in
Fitting
,, Stringer Plates .
Decks, Strength Value of Upper
Deep Tanks, Frame Riveting in Way of
„ Function of Centre Line

Bulkheads in .
„ Hatchways, . 120

,, Methods of obtaining water-

tightness at margin of
,, Usual Positions of

Derricks, Advantages of Plumb
,, Function of Cargo

,, Number of in Steamers
,, Value of Hydraulic .
Derrick Posts

Tables . . _ .

Displacement Calculation, Specimen

,, of Vessel out of Normal

Trim, To Obtain . 29s

Docking Stresses .... 7°

Doors in Watertight Bulkheads . . 171
Double Bottom, Advantages of a Contin-
uous . . in
,, Butts of Centre and Side

Girders in . . 116

,, Cellular System of Con-

structing . . 112, 115
,, Connection of Side Fram-

ing to margin of . 115. 116
,, Gusset Plates and Angles

at margin of . . 1 16

Partial ill

Plating of .116

,, Reduction in Thickness

of Shell Plating in way of 1 15
,, Riveted Connections of . 1 16

,, Strengthening Forward

in Full Vessels in way of 117
,, Testing of . . 120

Drift Punch, Objections to Excessive

Use of . . . . . . 96

Dynamical Stability, Definition of . . 245
„ Moseley's Formula for 24S
Ellipsoid, Volume of ... 14
Equilibrium of Floating Bodies, Condi-
tion of 177

Erections on Steamers, Structural Value of 77, 79
Five-Fight Rule for Areas . 8

Flanging of Plates in Lieu of Fitting

Ang!e Bars . . . 112

Flat-plate Keel, Description of . 96

Flat-plate Keel, with Intercostal Centre

Keelson . ... 96

Flat-plate Keel, with Centre through

Plate Keelson- .... 96
Floors, Connection to Centre Keelson 103

Floors, Ordinary .... 103

Form of Modern Cargo Steamers, De-
velopment of . . . . . 76
Frames, Advantages and Disadvantages

of joggled . . . 132

,, Reversed ... 42, 100, 101

,, Transverse . 42, 100

Frame Heel Pieces . . . 101

Frame Slips, Use of . 129, 130

Freeboard, Influence of on Curve of

Stability . . . . _ . 237-241

Garboard Strakes Precaution in Arrang-
ing End [oints of ... 128
Girders Under Deck . .123
Grain Cargoes, Causes of Shifting of . 279
,, Government Regulations

Regarding . 280

,, Margin of Stability Re-

quired with . 284

,, Stowage of . 278,308

Gudgeons, Rudder . 176/- 1 76/

Gusset Plates and Angles at Margin of

Double Bottoms . .116

Gyroscopic Apparatus for Minimising

Rolling . . . 267, 268

Hatch Cleats, spacing of . 152

Hatchways, cargo .... 14S-152

Hatch Coamings, Advantages of Round

Corners at Upper Deck . . 15°

Hatch Coamings, Scantlings of 149

Hatchways into Deep Tanks . 120-152

151

151
101

105

84
125

274
67

2/3

-190

236

95
89-92

2 57
163
128

Hatch Webs, Number of

,, ,, Method of fitting

Heel Pieces, Frame
Hogging and Sagging Strains.
Holds, Penalty exacted for Unobstructed
Hold Pillars, Compensation for Omis-
sion of .
,, ,, Wide Spaced . . 84,

Homogeneous Cargoes, Loading of
Horizontal Shearing Stress . . 60.

Inclining and Rolling Experiments, Value

of .
Inclining Experiment . 1S7

Integrator, Mechanical .
Intercostal and Single Plate Centre Keel-
sons, Comparison of
Isherwood Type of Steamer
Isochronous Rolling
[ibboom ...

Joints of Shell Plating, Arrangement of 127
Joints of Shell Plating, Comparison of

Overlapped and Butted .
Joints of Shell Plating, Lloyd's Rules for
Joints of Shell Plating, Method of curing

Leaky 134

Joints of Shell Plating, Methods of form-
ing

[oints of Shell Plating, Scarphing of End
Keel, Flat Plate .

,, Scarpb of Solid Bar
,, Scarpb, Use of Tack Rivets in
,, Side Bar ...
,, Solid Bar
Keelson and Hold Stringer butt straps.

Method of Fitting . . . 104, 105

Knighthead Plate . . 163

Law of Archimedes . . iS

Liquid Cargoes . . , 274 27S

Lloyd's Numerals, how Derived . . 97-100

Lloyd's Rules, Definitions of Length,

Breadth, and Depth, as given in . 98

Lloyd's Rules for Breadth of Shell Plates 129
Lloyd's Rules for Camber of Deck Beams 106
Lloyd's Rules for Diameters of Rivets . 137

133
12S

136

129
136
93
93
93
95
94

INDEX.

3 2 7

PAGR

Lloyd's Rules for Joints of Shell Plating . 128
Lloyd's Rules for Number of Transverse

Watertight Bulkheads . . . 167

Lloyd's Rules for Position of Collision

Bulkhead .... 166

Lloyd's Rules for Riveting of Edge Seams

of Shell Plating in Large Vessels . 133
Lloyd's Rules for Seasoning of Pine Deck

Planking 148

Lloyd's Rules for Spacing of Beams . 106
Lloyd's Rules Regarding Diameter of

Masts of Steamships , . .164

Loading and Ballasting 272-2S7, 292, 293

Loading of General Cargoes . 272, 273

,, HLmogeneous Cargoes , 274

Local Stresses ..... 73

Locus of Centres of Buoyancy . . 36, 40

Longitudinal Metacentre . 19S, 199, 202

Strains .... 51-53

,, Strength of Shallow Vessels 100

,, Stresses . . . 66, 67

Machinery Casings . . 146, 171

Masts of Steamships, Diameters of. . 164

Masts of Steamships, Staying of . , 165

Masts, Function of in Sailing-ships . 159

,, Function of in Steamers 156, 159, 164

,, Number of Plates in Round . 160

,, of Sailing-ships, Riveting of End

and Edge Joints in . . 160

,, Stresses on 161, 164

Mast Mountings, Importance of Strong , 164

Mast Steps, Construction of . . 161, 165

,, Wedging, Method of Fitting . . 161

Materials of Construction, Modern System

of Distributing 79

Mauretania, Longitudinal Stress on

R.M.S 66

M'Intyre System of Constructing Ballast

Tanks . . ..Ill

Mean Draught . . .20, 22, 293

Metacentre, Proof of Formula for Posi-
tion of . . . 299, 300
Transverse, Approximate Meth-
ods of finding Position of 183, 1S4
, , Transverse, Calculation for Posi-
tion of 181, 182
,, Transverse, Definition of . 179
Metacentric Height in Sailing-ships . 194
,, Height, Safe Minimum Value

of 194

,, Height, Transverse . 179

Metacentre, Longitudinal, in Vessels of

Simp'e Forms . 200

,, Longitudinal, in Vessels of

Ordinary Forms 198, 199

Metacentric Stability . .179

Moment of a Force . . . . 25, 26

,, of Inertia, Explanation of Term 180

,, of Inertia of a Waterplane 1S0, 181,

200, 201
,, of Inertia of a Section of a Beam 59
,, of Inertia of a Section of a Ship 65

,, of Stresses resisting bending of

Beams and Ships . . . 59, 65

,, to Alter Trim One Inch . 199, 213

Neutral Axis of a Beam . ... 58

PAGE

Neutral Axis of a Ship .... 62-64

,, Stress on Shell Plating at 69, 126

Normand's Approximate Trim Formula . 213

Oil Vessels, Bulkheads of 169

,, Isher wood's System Applied

to Construction of . , 91
,, Loading of . . . 277, 278

Outer Bottom, Function of . . . 125
,, Plating, Relative Value of

Different Parts of . 126

Outlines of Construction . . 42

Panting Strains ..... 73

Partial Awning Deck Type of Cargo

Steamer . ... 79

Peaks, Spacing of Transverse Frames in 100

Peak Tank, Bulkheads, Stiffening of . 117

Peak Tanks, Function of . . .117

,, Pitch of Shell Rivets thro'

Frames in way of 117

,, Testing of . . . . 120

,, Value of Wash Plates in . 117

Period of Roll 256

, , Effect of Motion Ahead on 266

Period of Wave ..... 258
Pillars, Arrangement of, for Shifting

Boards . . . .122

,, Comparison of Short and Long 120

,, Heads and Heels of . . 121- 124

„ Number of Rows Required . 121

,, Quarter . . . 121

Pillars, in Deep Tanks . . . .119

,, Portable .... 123, 124

,, Rivets in End Attachments of . 123

,, Runners under Beams for . . 122

,, Wide Spaced . . 124, 125

Pintle, Detail of Bottom . . 176/, 1767'

,, Function of Lock . 176/*

Pitching and Heaving . . 268-270

Pounding Strains . . . J^ y 285

Prismatic Co-efficient .... 303

Propeller Brackets for Twin Screw

Steamers .... 176^, 176^

Pyramid, Volume of a . . . . 14

Quarter-deck Type of Steamer . . 78

Quarter-deck Type of Steamer, Compen-
sation at Break of Main Deck in . 78
Radius of Gyration, Transverse . . 256
Rates of Stowage . . . 308
Rectangle, Area of ... 2
Resistance of Beams to Change of Form . 56
Reversed Frames . . 42, 100
Rhomboid, Area of ... 2
Righting Moments by Metacentre Method 217
,, Curve of . 233
Rigging Screws, Use of ... 163
Rivet Holes, Advantages and Disadvan-
tages of Drilling . . 140
,, Method of Correcting Blind

and Partially Blind . 141

,, Objections to Punching . 14O

Riveted Connections, Strength of , 141

,, ,, of Stern Post to

Shell Plating 176a, 176^
,, Joints, Experiments to find

Strength of . , . . 141

,, Joints, Frictional Strength of . 141

3 23

INDEX.

Riveting of Edge Seams of Shell Plating

in Large Vessels ....

Riveting of Edge and End Joints of

Masts . . . i6o, 164

,, Edge and End Joints of

Top Masts
Edges of Sheer Strakes
End Joints of Bowsprit
Plating ....
Frames to Shell Plating in

way of Deep Tanks 119, 138

Joints of Watertight Bulk-
heads .... 168
Shell Plating of R.M.S.
Lusj'tam'a and Mauretania
Stem to Shell Plating 175,
Rivets, Considerations Governing Sizes of
Forms of Heads and Points of 13S.
for Watertight Wo-k, Spacing of 137
in Bulkhead Frames, Spacing of
in End Attachments of Pillars
in Flat Plate Keels, Spacing of .
in Joints of Shell Plating, Spacing of
in Seams of Shell Plating, number

of Rows of
Spacing of in Bar Keels . 94,

Strength in Iron Plates of Iron .
,, Steel Plates of Iron .

Through Frames and Shell Plat-
ing in Oil Compartments of
Bulk Oil Vessels .
Rivet Holes, Countersinking of .

,, Precaution Necessary in

Marking and Punching

Rolling of Ships, Analysis of Resistance to

,, Effect of Synchronism of

Periods of Ship and

Waves on . . 261, 263

,, Experiments with Water

Chambers . . 266, 267
,, Extinctive Value of Bilge

Keels . . 264-266

,, Influence of Change of

Course and Speed on .
,, Influence of Metacentric

Height on .
,, Instantaneous Axis of

Rotation . . 254,

,, Isochronous .

,, Resistance to

,, Use of Gyroscope for

Minimising
Rudder, Alternative Plans for Carrying
Weight of
Balanced, advantage of a .
Bow .....

Coupling. . . . 176/, 176;;/
Consisting of Forged Frame and

Side Plates . . . 1 jbm- 1760
Gudgeons .... iy6r

Considerations Governing Scant-
lings of .... 176c
of Large Steamer . , 176^/, 176/
Pintles .... 176/-176/
Relative Meritsjof Cast Steel and

Forgings for , , T . 176.-

133

162
133

163

137
176
137
139
,138
138
123

I3S
138

133

I3S
141

141

138
139

140
264

263
256

255
257
263

267

176/
176/
176/

PAGE

Rudder, Single Plate . . . 1762, 176/

,, Stops . ... 176/&

,, Bearing or Thrust Block . . 176/
Sailing-ships, Watertight Bulkheads in . 167
Sails in Steamships, use of . . 164

Scale of Deadweight .... 21
Scantling Numbers, Lloyd's . . 97-100

Scarph, Bar Keel 93

Scarphing of End Joints of Shell-plating

in Way of Seams . . . 135, 136

Sea Waves, Theory of . . 257

Sections of Beams . . . 107

Self-trimming Types of Cargo Steamers 86-89
Shaft Bossing in Twin Screw Steamers 176^, 176^
Shallow Vessels, Provision for Strength-
ening . . . . . .100

Shearing Forces ..... 45-56

,, Graphical Method of

Finding . . 49, 51, 56
Sheerstrakes, Precaution in Arranging

End Joints of . 128

,, Riveting of Edge Seams of 133

Shear Stress, Maximum. . . 68, 69

,, Mean .... 67

Shearing Stresses, Position of Maximum

Longitudinal .... 69, 127

Shell Plating, Advantages and Disadvant-
ages of Joggled . . 131
,, Advantages and Disadvant-

ages of butted End Joints 133
,, Arrangement of End Joints

of. 127, 128

Methods of Forming Joints
of . 129

,, of Small Vessels at Ends,

Taper of . . 127

,, Precautions necessary in

working . . . 136

, , Reason of Comparative

Uniformity in Thickness

of 127

,, Relative Importance of Dif-

ferent Paris of . .126

,, Riveting of Edge Seams in

Large Vessels . . 133

,, Scantlings of in Two Cases 127

,, Thickness at Sternpost of . 127

Shell Plates, Advantages and Disadvant-

ofWide. . . . 129

,, Lloyd's Rules for Breadths of 129

Shelter Deck Type of Steamer . . 76, 77

Shift of Cargo, Effect of . . 2S2, 283

Shrouds, Mast . 162

Side Bar Keel . . ge 96

Side Keelsons . ... 103

Side Stringers, Connections to Bulk-
heads . . 171, 172
»> •>* Function of . . 82, 104
»» ^ How Constructed . . 104
»* >< Number Required by

Lloyd's Rules . . 104

Simpson's Rules, Application of 6-10, 16, 17,
19, 30. 32. 35i 40. 1S2, 202, 22S, 247
Single Deck Vessels, Size of Beam Knees

in Large I0g

Single Plate Rudder , , 176^ 176/, 176/

INDEX.

3-9

C 1 ,r PAG "'

Sphere, Volume of ... 14

Square, Area of . I

Stability, Dynamical 245

,, Effect of a Squall on . 249-251

Information for Commanding

Officers ' 288

,, Initial . 187

,, Metacentric . .179

Statical, Atwood's Formula for 221

Statical, Causes Influencing

Forms of Curves . 237

Statical, Cross Curves ot . 231-236

,, Statical, Effect of Adding or

Removing Weights . 193, 194
,, Statical, Effect of Consumption

of Bunker Coal . 192, 284

,, Statical, Safe Curve of 244

Statical, Specimen Calculation

228-230
,, Statical, Tangent to Curve at

Origin 230, 231

Standing Rigging , 162, 165

Steadiness . . 272

Steam Winches, Arrangement for Sup-
porting . 157
,, Arrangement of . 157
,, Seats for . 15S
Steam Winch Pipe Stools . . .159
Stem, Connection to Bar Keel . 173, 174
,, ., Flat Plate Keel 174, 175
Side Bar Keel 174, 175
Ordinary Form of . 173
Scarphs, Position of .176
Sternpost, Scarphs of . 176c, 176^
Sternposts of Single Screw Steamers Ij6a-ij6d
Siernposts of Twin Screw Steamers 176/
Sternposts, Relative Merits of Steel Cast-
ings and Forgings for . . 176^
Sternposts, Connection to Shell Plating

176.2:, 176^
Sternposts of Sailing Ships 176

Stowage Rates . . . 308

Strength of Beams, Influence of Form of

Section on .61

Strength Types of Cargo Steamers . 75

Stress at any Point of a Section of a Ship 64
Stresses, Compressive, Effect of on thin

Deck-plating ... 66

Due to Action of Propeller 73

Due to Docking . 70

,, on Shell Plating

, , ' Position of Maximum

tudinal Bending .
,, Position of Maximum

tudinal Sheering .
,, Transverse, Due to Rolling
Stringer Plates on Deck Beams
Submarine Vessels, Stability of

Longi-

. 66, 69

Longi-

66, 126

67,

126
72

143
219, 220

Tables of Natural Tangents, Sines and

Cosines 3°5-3°7

Tack Rivets, Function of . .176

(J Objection to use of . 93, 176

Tap Rivets .... 140

TchebychefTs Rules for Areas . 11, 12

Telescopic Topmasts . . . .164

Three Island Type of Cargo Steamer . 77

'AGP,

M3
281

Tieplates, Deck

Timber Cargoes, Stowage of .

,, Type of Vessel Suitable

for . . . 28 1

Topmasts, Method of Fitting . 161, 164

,, Riveting of Joints of plating of 162

Tons per Inch Immersion . 22

Tranaverse Frames, Spacing of 100

1, ,, Types of. . . 100

Transverse Metacentre . 179, 1S1, 1^2

, , M eta ccn tre, Approximate

Methods of Finding Posi-
tion of . . 183, 184
,, Metacentres, Curve of . 185, 1S6
,, Metacentric Height, Safe

Minimum Value of
, , Metacentric Height in Sailing-

ships
,, Stresses Due to Incorrect

Loading
Stresses Due to Rolling
,, System of Construction

Trapezoid, Area of a
Trapezoidal Rule for Curvilinear Areas
Triangle, Area of a
Trim, Approximate Calculation of .
Change of .
Cross Curves of .
Effect of Filling Fore-peak Tank on
In formation for Commanding

Officers ...
Lines . . . 207,

Mr. Long's Method . 206

Normand's Approximate Formula
Worked-out Examples 203, 205,

Trochoidal Theory of Waves 257,

Tunnel Door . . . . 171

Type of Steamer, Awning 01 Shelter Deck 76, 77
,, Dixon <N: Harroway . 87, 88

,, Isherwood .

,, Partial Awning Deck

Quarter-deck
,, Ropner Trunk

,, Single Deck

,, Three Island

, , Turret

Well Deck .
Twin Screw Steamer Propeller Bracket

194

7 2

42-44

213, 214

19S, 213

209, 211

205

^5
208
209

25S
173

89-92
79

7*
87
S3

77
86

170?
Volume of Ellipsoid . 14

,, of Pyramid . . 14

, , of Rectangular, Solid . . 14

of Ship ' . 14, 15, 16

,, of Sphere . ... 14

,, of Wedge bounded by Curved

Surface . 221, 222, 224

,, Units of 13

Water Ballast, Advantage bf High Posi-
tion of .120
,, Chambers . . 266, 267
Watertight Bulkhead Doors into 'Tween

Decks . . . 171. 174

,, Bulkhead Doors, Arrange-

ments for Operating 172, 173

Watertight Bulkheads . . 165-171

Wave, Period of . 25S

33°

INDEX.

]'AGB

Wave, Speed of .... 258
Waves, Lengths and Periods of Atlantic

Storm 258, 259

Web Frame, Connections Lo Beams and

Inner Bottom 103
,, Definition of . 102
Web Frames and Side Stringers, Compar-
ative Strength of . . 103
Weight of Fresh Water . . 297
Weights of Materials 307
Weight of Salt Water 18

l'AGB

Wind Curve . . . 249, 250

Wood Decks, Comparative Values of

Different Timbers for 147
,, Fastenings for. . 147

,, Objections to Use of Slips

in Laying . . 148

,, Precautions necessary in

laying . . .146

, , Seasoning of Planking

intended for 147

Yards . 163

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Full text of "Ship construction and calculations, with numerous illustrations and examples for the use of officers of the mercantile marine, ship superintendents, draughtsmen, etc."

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